hep-th/0109133 Gauss-Bonnet Black Holes in AdS Spaces ∗ Rong-Gen Cai Institute of Theoretical Physics, Chinese Academy of Sciences, P.O. Box 2735, Beijing 100080, China Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan 2 0 0 Abstract 2 n a Westudythermodynamicpropertiesandphasestructuresoftopologicalblack J holes in Einstein theory with a Gauss-Bonnet term and a negative cosmolog- 2 1 ical constant. The event horizon of these topological black holes can be a hypersurface with positive, zero or negative constant curvature. When the 2 v horizon is a zero curvature hypersurface, the thermodynamic properties of 3 blackholesarecompletely thesameasthoseofblack holeswithouttheGauss- 3 1 Bonnet term, although the two black hole solutions are quite different. When 9 thehorizonisanegativeconstantcurvaturehypersurface,thethermodynamic 0 1 properties of the Gauss-Bonnet black holes are qualitatively similar to those 0 of black holes without the Gauss-Bonnet term. When the event horizon is / h a hypersurface with positive constant curvature, we find that the thermody- t - namicpropertiesandphasestructuresofblackholesdrasticallydependonthe p e spacetime dimension d and the coefficient of the Gauss-Bonnet term: when h d 6, the properties of black hole are also qualitatively similar to the case : ≥ v withouttheGauss-Bonnetterm,butwhend =5,anewphaseoflocally stable i X small black hole occurs under a critical value of the Gauss-Bonnet coefficient, r and beyond the critical value, the black holes are always thermodynamically a stable. However, the locally stable small black hole is not globally preferred, insteadathermalanti-deSitterspaceisglobally preferred. Wefindthatthere is a minimal horizon radius, below which the Hawking-Page phase transition will not occur since for these black holes the thermal anti de Sitter space is always globally preferred. ∗ Email address: [email protected] 1 I. INTRODUCTION In recent years black holes in anti-de Sitter (AdS) spaces have attracted a great deal of attention. There are at least two reasons responsible for this. First, in the spirit of AdS/CFT correspondence [1–3], it has been convincingly argued by Witten [4] that the thermodynamics of black holes in AdS spaces (AdS black holes) can be identified with that of a certain dual CFT in the high temperature limit. With this correspondence, one can gain some insights into thermodynamic properties and phase structures of strong ’t Hooft coupling CFTs by studying thermodynamics of AdS black holes. Second, although the “topological censorship theorem” [5] still holds in asymptotically AdSspaces [6], it hasbeenfoundthat except for thespherically symmetric black holeswhose event horizon is a sphere surface, black holes also exist with even horizon being a zero or negative constant curvature hypersurface. These black holes are referred to as topological blackholesintheliterature. Duetothedifferent horizonstructures, theseblack holesbehave in many aspects quite different from the spherically symmetric black holes [7]- [19]. It is by now known that the AdS Schwarzschild black hole is thermodynamically un- stable when the horizon radius is small, while it is stable for large radius; there is a phase transition, named Hawking-Page phase transition [20], between the large stable black hole and a thermal AdS space. This phase transition is explained by Witten [4] as the confine- ment/deconfinement transition of the Yang-Mills theory in the AdS/CFT correspondence. However, it is interesting to note that if event horizon of AdS black holes is a hypersurface with zero or negative constant curvature, the black hole is always stable and the corre- sponding CFT is always dominated by the black hole. That is, there does not exist the Hawking-Page phase transition for AdS black holes with a Ricci flat or hyperbolic hori- zon [16]. Higher derivative curvature terms occur in many occasions, such as in the semiclassi- cally quantum gravity and in the effective low-energy action of superstring theories. In the latter case, according to the AdS/CFT correspondence, these terms can be viewed as the corrections of large N expansion of boundary CFTs in the strong coupling limit. Due to the non-linearity of Einstein equations, however, it is very difficult to find out nontrivial exact analytical solutions of the Einstein equations with these higher derivative terms. In most cases, one has to adopt some approximation methods or find solutions numerically. Amongthegravitytheorieswithhigherderivativecurvatureterms, theso-calledLovelock gravity [21] is of some special features in some sense. For example, the resulting field equations contain no more than second derivatives of the metric and it has been proven to be free of ghosts when expanding about the flat space, evading any problems with unitarity. The Lagrangian of Lovelock theory is the sum of dimensionally extended Euler densities1 n = c , (1.1) i i L L i X where c is an arbitrary constant and is the Euler density of a 2i-dimensional manifold, i i L 1The gravity theory with a Gauss-Bonnet term was originally proposed by Lanczos [22], indepen- dently rediscovered by Lovelock [21]. See also discussions in [23] and [24]. 2 = 2−iδa1b1···aibiRc1d1 Rcidi . (1.2) Li c1d1···cidi a1b1 ··· aibi Here = 1 and hence c is just the cosmological constant. gives us the usual Einstein- 0 0 1 L L Hilbert term and is the Gauss-Bonnet term. A spherically symmetric static solution of 2 L (1.1) has been found in the sense that the metric function is determined by solving for the real roots of a polynomial equation [25]. Since the Lagrangian (1.1) includes many arbitrary coefficients c ,itisdifficult toextractphysicalinformationfromthesolution. InRefs.[26,27], i by restricting these coefficients to a special set so that the metric function can be readily determined by solving the polynomial equation, some exact, spherically symmetric black hole solutions have been found. Black hole solutions with nontrivial topology in this theory have been also studied in Refs. [17,19]. In this paper we will analyze black hole solutions in Einstein theory with a Gauss-Bonnet term and a negative cosmological constant, in which the Gauss-Bonnet coefficient is not fixed. In this theory a static, spherically symmetric black hole solution was first discovered by Boulware and Deser [28]. However, the thermodynamic properties of the solution were not discussed there. Here we will first generalize this solution to the case in which the horizon of black holes can be a positive, negative or zero constant curvature hypersurface, and then discuss thermodynamic properties and phase structures of black holes. Because of this Gauss-Bonnet term, some nontrivial and interesting features will occur. II. TOPOLOGICAL GAUSS-BONNET BLACK HOLES The Einstein-Hilbert action with a Gauss-Bonnet term and a negative cosmological con- stant, Λ = (d 1)(d 2)/2l2, in d dimensions can be written down as [28] 2 − − − 1 (d 1)(d 2) S = ddx√ g R+ − − +α(R Rµνγδ 4R Rµν +R2) , (2.1) 16πG − l2 µνγδ − µν ! Z where α is the Gauss-Bonnet coefficient with dimension (length)2 and is positive in the heterotic string theory [28]. So we restrict ourselves to the case α 0 3. Varying the action ≥ yields the equations of gravitational field 1 (d 1)(d 2) 1 R g R = − − g +α g (R Rγδλσ 4R Rγδ +R2) µν − 2 µν 2l2 µν 2 µν γδλσ − γδ (cid:18) 2RR +4R Rγ +4R Rγ δ 2R R γδλ . (2.2) − µν µγ ν γδ µ ν − µγδλ ν (cid:17) We assume the metric being of the following form ds2 = e2νdt2 +e2λdr2 +r2h dxidxj, (2.3) ij − 2The Gauss-Bonnet term is a topological invariant in four dimensions. So d 5 is assumed in ≥ this paper. 3We will make a simple discussion for the case α< 0 in Sec. III. 3 where ν and λ are functions of r only, and h dxidxj represents the line element of a (d 2)- ij − dimensional hypersurface with constant curvature (d 2)(d 3)k and volume Σ . Without k − − loss of the generality, one may take k = 1, 1 or 0. Following Ref. [28] and substituting the − ansatz (2.3) into the action (2.1), we obtain (d 2)Σ rd−1 ′ S = − k dt dreν+λ rd−1ϕ(1+α˜ϕ)+ , (2.4) 16πG " l2 # Z whereaprimedenotesderivativewithrespect tor,α˜ = α(d 3)(d 4)andϕ = r−2(k e−2λ). − − − From the action (2.4), one can find the solution eν+λ = 1, 1 16πGM ϕ(1+α˜ϕ)+ = , (2.5) l2 (d 2)Σ rd−1 k − from which we obtain the exact solution4 r2 64πGα˜M 4α˜ e2ν = e−2λ = k + 1 1+ , (2.6) 2α˜ ∓s (d 2)Σkrd−1 − l2 ! − where M is the gravitational mass of the solution5. The solution with k = 1 and spherical symmetry was first found by Boulware and Deser [28]. Here we extend this solution to include the cases k = 0 and 1. Note that the solution (2.6) has two branches with “ ” − − or “+” sign. Moreover, there is a potential singularity at the place where the square root vanishes in (2.6), except for the singularity at r = 0. Here we mention that the theory (2.1) with α˜ = l2/4 discussed in Ref. [27] in five dimensions corresponds to the theory proposed in [30]; the solution with α˜ = l2/4 discussed in Refs. [26,27] in five dimensions was also included in Refs. [17,19]. If α˜ = 0, namely, without the Gauss-Bonnet term, the solution (2.6) reduces to the one in [16], and the thermodynamics of the latter was discussed there. When M = 0, the vacuum solution in (2.6) is r2 4α˜ e−2λ = k + 1 1 . (2.7) 2α˜  ∓s − l2    Since α˜ > 0, one can see from the above that α˜ must obey 4α˜/l2 1, beyond which this ≤ theory is undefined. Thus, the action (2.1) has two AdS solutions with effective cosmological constants l2 = l2 1 1 4α˜ . When 4α˜/l2 = 1, these two solutions coincide with each eff 2 ± − l2 (cid:16) q (cid:17) 4It is not so obvious that the minisuperspace approach applies for non-spherically symmetric solutions in the gravity theory. However, it can be checked that the solution (2.6) indeed satisfies the equations (2.2) of motion. This is related to the fact that following [29], one can show that a Birkhoff-like theorem holds in the gravity theory (2.1). 5This gravitational mass can be obtained by substituting the solution (2.6) into the action (2.4) and then using boundary term method. For this method, for example, see [27]. 4 other, resulting in e−2λ = k+2r2/l2 and that the theory has a unique AdS vacuum [26,27]. On the other hand, if α˜ < 0, the solution (2.7) is still an AdS space if one takes the ” ” sign, − but becomes a de Sitter space if one takes the ”+” sign and k = 1. From the vacuum case, the solution (2.7) with both signs seems reasonable, from which we cannot determine which sign in (2.6) should be adopted. This problem can be solved by considering the propagation of gravitons on the background (2.7). It has been shown by Boulware and Deser [28] that the branch with “+” sign is unstable and the graviton is a ghost, while the branch with “ ” sign is stable and is free of ghosts. This can also be seen from the case M = 0. − 6 When k = 1 and 1/l2 = 0, just as observed by Boulware and Deser [28], the solution is asymptotically a Schwarzschild solution if one takes the “ ” sign, but is asymptotically an − AdS Schwarzschild solution with a negative gravitational mass for the “+” sign, indicating the instability. Therefore the branch with “+” sign in(2.6)is of less physical interest6. From now on, we will not consider the branch with “+” sign. From (2.6), the mass of black holes can be expressed in terms of the horizon radius r , + (d 2)Σ rd−3 α˜k2 r2 k + + M = − k + + . (2.8) 16πG r+2 l2 ! The Hawking temperature of the black holes can be easily obtained by requiring the absence of conical singularity at the horizon in the Euclidean sector of the black hole solution. It is 1 ′ (d 1)r4 +(d 3)kl2r2 +(d 5)α˜k2l2 T = e−2λ = − + − + − . (2.9) 4π 4πl2r (r2 +2α˜k) (cid:16) (cid:17)(cid:12)(cid:12)r=r+ + + (cid:12) Usually entropy of black h(cid:12)oles satisfies the so-called area formula. This is, the black hole entropy equals to one-quarter of horizon area. In gravity theories with higher derivative curvature terms, however, in general the entropy of black holes does not satisfy the area formula. To get the black hole entropy, in [17] we suggested a simple method according to the fact that as a thermodynamic system, the entropy of black hole must obey the first law of black hole thermodynamics: dM = TdS. Integrating the first law, we have −1 r+ −1 ∂M S = T dM = T dr , (2.10) + Z Z0 ∂r+! where we have imposed the physical assumption that the entropy vanishes when the horizon of black holes shrinks to zero7. Thus once given the temperature and mass of black holes in terms of the horizon radius, One can readily get the entropy of black holes and needs not 6A detailed analysis of the solution (2.6) without the negative cosmological constant, namely, 1/l2 =0, has been made in [32,33]. 7Note that for thek = 1black hole, thereexists aminimalhorizon radius. For these black holes, − therefore the horizon cannot shrink to zero. However, it is known that the black hole entropy is a functionofhorizonsurface[31]. Accordingtothesecondlawofblackholemechanics, theblackhole entropy can be expressed in terms of a polynomial of horizon radius r+ with positive exponents. As a result, although the black hole horizon cannot shrink to zero when k = 1, this method − 5 know in which gravitational theory the black hole solutions are. Substituting (2.8) and (2.9) into (2.10), we find the entropy of the Gauss-Bonnet black holes (2.6) is Σ rd−2 (d 2)2α˜k k + S = 1+ − . (2.11) 4G (d 4) r+2 ! − When k = 1, it is in complete agreement with the one in [32], there the entropy of the Gauss-Bonnet black holes without the cosmological constant is obtained by calculating the Euclidean action of black holes. The heat capacity of black holes is ∂M ∂M ∂r + C = = , (2.12) ∂T ! ∂r+! ∂T ! where ∂M (d 2)Σ = − krd−5(r2 +2α˜k)T, + + ∂r 4G + ∂T 1 6 2 4 4 = (d 1)r (d 3)kl r +6(d 1)kα˜r ∂r 4πl2r2(r2 +2α˜k)2 − + − − + − + + + + h 2 2 2 2 2 2 2 2 +2(d 3)α˜k l r 3(d 5)α˜kl r 2(d 5)α˜ k l ]. (2.13) + + − − − − − The free energy of black holes, defined as F = M TS, is − Σ rd−5 k + 6 2 4 F = (d 4)r +(d 4)kl r 16πG(d 4)l2(r2 +2α˜k) − − + − + + − h 4 2 2 2 2 2 6(d 2)kα˜r +(d 8)α˜k l r +2(d 2)α˜ kl ]. (2.14) + + − − − − Thus we give some thermodynamic quantities of Gauss-Bonnet black holes in AdS spaces. When α˜ = 0, these thermodynamic quantities reduce to corresponding ones in Ref. [16]. In Fig. 1 the inverse temperature β = 1/T of the black holes versus the horizon radius is plotted. We can see clearly different behaviors for the cases k = 1, 0 and 1: The inverse − temperature always starts from infinity and monotonically decreases to zero in the cases k = 0 and k = 1, while it starts from zero and reaches its maximum at a certain horizon − radius and then goes to zero monotonically when k = 1. This indicates that for the cases k = 1 and k = 0, the black holes are not only locally thermodynamic stable, but also − globally preferred, while in the case of k = 1, the black hole is not locally thermodynamic stable for small radius, but it is for large radius. Therefore, for the k = 1 case, there is a Hawking-Page phase transition. For details see [16]. When α˜ = 0, we see that those quantities drastically depend on the parameter α˜, horizon 6 structure k and the spacetime dimension d. Below we will discuss each case according to the classification of horizon structures, k = 0, k = 1 and k = 1, respectively. − seems applicable as well. The results in [17] and in this paper show this point. For example, when α˜ = 0, the formula (2.11) gives the entropy of AdS black holes in Einstein theory without the Gauss-Bonnet term. Obviously, in this case the resulting area formula (2.11) holds as well in the case of k = 1. − 6 A. The case of k = 0 In this case we have (d 1)r + T = − , 4πl2 Σ S = krd−2, + 4G (d 2)Σ C = − krd−2, + 4G Σ rd−1 k + F = , (2.15) −16πG l2 where rd−1 = 16πGl2M/(d 2)Σ . It is interesting to note that these thermodynamic + k − quantities are independent of the parameter α˜. That is, these quantities have the completely same expressions as those [16] for black holes without the Gauss-Bonnet term. We therefore conclude that in the case k = 0, the black holes with and without Gauss-Bonnet term have completely same thermodynamic properties, although the two solutions are quite different, which can be seen from (2.6). In particular, we note here that the entropy of the Gauss- Bonnet black holes still satisfies the area formula in the case k = 0. B. The case of k = 1 − As the case [16]without the Gauss-Bonnet term, there arealso so-called “massless” black hole and “negative” mass black hole in the Gauss-Bonnet black hole (2.6). When M = 0, the black hole has the horizon radius l2 4α˜ 2 r = 1 1 , (2.16) + 2  ±s − l2    with Hawking temperature T = 1/2πr . Here there are two “massless” black hole solutions, + corresponding to two branches in the solution (2.6). But the black hole with smaller horizon radius belongs to the unstable branch. Given a fixed α˜, the smallest black hole has the horizon radius (d 3)l2 (d 1)(d 5)4α˜ 2 r = − 1+ 1 − − . (2.17) min 2(d 1)  v − (d 3)2 l2  u − u −  t  The black hole is an extremal one, it has vanishing Hawking temperature and the most “negative” mass (d 2)(d 3)Σ l2rd−5 d 14α˜ (d 1)(d 5)4α˜ k min M = − − 1 − + 1 − − . (2.18) ext − 16πG(d 1)2  − d 3 l2 v − (d 3)2 l2  u − − u −  t  When 4α˜/l2 = 1, the smallest radius is r2 = l2/2 and M = 0, independent of the min ext spacetime dimension d. But in this case, the Hawking temperature does not vanish. It is T = 1/√2πl. This is an exceptional case. 7 From the solution (2.6), one can find that in order for the solution to have a black hole horizon, the horizon radius must obey 2 r 2α˜. (2.19) + ≥ Thus the smallest radius (2.17) gives a constraint on the allowed value of the parameter α˜: 2 r 2α˜, (2.20) min ≥ which leads to 4α˜/l2 1. Since the theory is defined in the region 4α˜/l2 1, the condition ≤ ≤ (2.20) is always satisfied. Due to the existence of the smallest black holes (2.17), we see from (2.9) that except for the case 4α˜/l2 = 1, the temperature of black hole always starts from zero at the smallest radii, corresponding to the extremal black holes and monotonically goes to infinity as r . In the case 4α˜/l2 = 1, the temperature starts from 1/√2πl at + → ∞ r2 = l2/2. This can also be verified by looking at the behavior of the heat capacity (2.12). + After considering the fact that r2 2α˜ and 4α˜/l2 1, it is easy to show that the heat + ≥ ≤ capacity is always positive. In Fig. 2 we plot the inverse temperature of black holes in six dimensions versus the parameter α˜/l2 and the horizon radius r /l. + Amongthesmallestblackholes(2.17),themostsmallestoneisr2 = l2/2when4α˜/l2 = 1, + its free energy is zero. Therefore the free energy is always negative for other black holes since the heat capacity is always positive. As a result, the thermodynamic properties of the black holes with the Gauss-Bonnet term are qualitatively similar to those of black holes without the Gauss-Bonnet term: These black holes are always stable not only locally, but also globally. In addition, let us note that except for the singularity at r = 0, the black hole solution (2.6) has another singularity at 4α˜rd−3 α˜ r2 rd−1 = + 1 + , (2.21) s 1 4α˜/l2 − r+2 − l2 ! − when M < M < 0 . But both singularities are shielded by the event horizon r . ext + C. The case of k = 1 This case is very interesting. From the temperature (2.9) one can see that the case d = 5 is quite different from the other cases d 6. When d = 5, the temperature starts from ≥ zero at r = 0 and goes to infinity as r , while it starts from infinity at r = 0 + + + → ∞ as d 6. In Fig. 3 we show the inverse temperatures of black holes with α˜/l2 = 0.001 in ≥ different dimensions d = 5, 6 and d = 10, respectively. The behavior of temperature of black holes with the Gauss-Bonnet term in d 6 dimensions is similar to that of AdS black holes ≥ without the Gauss-Bonnet term. But the case of d = 5 (see Fig. 4) is quite different from the corresponding one without the Gauss-Bonnet term (see Fig. 1). Comparing Fig. 4 with Fig. 1, we see that a new phase of stably small black hole occurs in the Gauss-Bonnet black holes. When d = 5, we have from (2.8) the black hole horizon 8 l2 4(m α˜) 2 r = 1+ 1+ − , (2.22) + 2 − s l2    where m = 16πGM/3Σ . Therefore, in this case there is a mass gap M = 3Σ α˜/(16πG): k 0 k all black holes have a mass M M . Using the horizon radius, from Fig. 4 we can see that 0 ≥ the black holes can be classified to three branches: branch 1 : 0 < r < r , C > 0, + 1 branch 2 : r < r < r , C < 0, 1 + 2 branch 3 : r < r < , C > 0, (2.23) 2 + ∞ where l2 12α˜ 16α˜ 12α˜ −2 2 r = 1 1 1 1 . (2.24) 1,2 4 − l2  ∓s − l2 − l2  (cid:18) (cid:19) (cid:18) (cid:19)   with the assumption 36α˜/l2 < 1. In the branch 1 and 3, the heat capacity is positive, while it is negative in the branch 2. Therefore the black holes are locally stable in the branch 1 and 3, and unstable in the branch 2. At the joint points of branches, namely, r = r , the + 1,2 heat capacity diverges. Comparing with the case without the Gauss-Bonnet term, one can see that the branch 1 is new. When α˜ increases to the value, α˜/l2 = 1/36, we find that the branch 2 with negative heat capacity disappears. Beyond this value, the heat capacity is always positive and the Gauss- Bonnet black holes are always locally stable. In Fig. 5, we show the inverse temperatures of Gauss-Bonnet black hole with the parameter α˜/l2, subcritical value 0.001, critical value 1/36, and supercritical value 0.20, respectively. In Fig. 6, the continuous evolution of the inverse temperatureisplottedwiththeparameterα˜/l2 fromzero to0.25,fromwhichonecan see clearly that the black holes evolve from two branches to one branch via three branches. However, inspecting the free energy (2.14) reveals that these stably small black holes are not globally preferred: The free energy always starts from some positive value at r = 0 + and then goes to negative infinity as r . In Fig. 7 the free energy of black holes with + → ∞ different parameter α˜/l2 is plotted. We see that all curves cross the horizontal axis (horizon radius) one time only, where F = 0. In Fig. 8 we plot the region where the free energy is negative. The region is α˜ < α˜ < α˜ , (2.25) 1 2 where r2 3r4 r2 9r4 11r2 5 + + + + + α˜ = + + . (2.26) 2,1 4 2l2 ± 2 s l4 3l2 − 12 The joint point of the two curves is at α˜/l2 = 0.0360 and r /l = 0.3043. Beyond this region, + the thermal AdS space is globally preferred. We see that there is a smallest horizon radius r /l = 0.3043: there will not exist the Hawking-Page phase transition when the black hole + horizon is smaller than the value r /l = 0.3043. When black holes cross the curves α˜ and + 2 α˜ , a Hawking-Page phase transition happens. 1 9 The region in which black holes are locally stable is determined by the curve α˜ , 0 l2r2 2r4 + + α˜ = − . (2.27) 0 2l2 +12r2 + In Fig. 9 the curve α˜ is plotted (the lowest one): the region is locally stable above this 0 curve, namely, α˜ > α˜ , and locally unstable below this curve. This curve α˜ touches the 0 0 curve α˜ at α˜ = 1/36 0.0278 and r /l = 0.4082. Unfortunately, in Fig. 9 most part of the 1 + ≈ curve α˜ is outside the plot. In Fig. 9 one can see that there is a large region where black 2 holes are locally stable, but not globally preferred. When d 6, unlike the case d = 5, there is no the mass gap. The properties of Gauss- ≥ Bonnetblackholesarequalitatively similar tothoseofblackholeswithout theGauss-Bonnet term. ThiscanbeseenfromthebehavioroftheHawkingtemperatureofblackholesinFig.3. Thisimpliesthattheequation ∂T = 0hasonlyonepositiverealrootr = r (d,α˜/l2). Using + 0 ∂r+ (2.13), one can obtain the positive real root. But its expression is complicated, so we do not present it here. Given a spacetime dimension d and a fixed parameter α˜/l2, when a black hole has a horizon r > r , the black hole is locally stable. Otherwise, it is unstable. + 0 The free energy (2.14) always starts from zero in the case d 6, reaches a positive ≥ maximum at some r , and then goes to negative infinity as r . This behavior is the + + → ∞ same as the case without the Gauss-Bonnet term (see the curve of α˜ = 0 in Fig. 7). The region where the black hole is globally preferred is restricted by a relation like (2.25), but with r2 + 2 2 α˜ = 6(d 2)r (d 8)l 2,1 4(d 2)l2 − + − − − h 36(d 2)2r4 4(d 2)(d 16)l2r2 +d(32 7d)l4 ]. (2.28) + + ± − − − − − q And as in the case of d = 5, these two curves connect at l2 r2 = d 16+ (d 16)2 +9d(7d 32) , + 18(d 2) − − − − (cid:18) q (cid:19) r2 + 2 2 α˜ = 6(d 2)r (d 8)l . (2.29) 4(d 2)l2 − + − − − (cid:16) (cid:17) in the α˜ r plane. Therefore the phase structure of black holes in d 6 dimensions is + − ≥ similar to the one in d = 5 dimensions (Fig. 8). Finally let us mention that the temperature behavior (Fig. 4) of d = 5 Gauss-Bonnet black holes is quite similar to the one of the Reissner-Nordstr¨om (RN) black holes in AdS spaces in the canonical ensemble [34,35]. There under the critical value of charge, the phase of stably small black holes occurs as well. However, there is a big difference between two cases: For the RN black holes, the small black hole is not only locally stable, but also globally preferred, while the small Gauss-Bonnet black hole is only locally stable and not globally preferred, instead a thermal AdS space is preferred. III. CONCLUSIONS AND DISCUSSIONS Wehave presented exact topologicalblackholesolutionsinEinstein theorywithaGauss- Bonnet term and a negative cosmological constant, generalizing the spherically symmetric 10