Gauss-Bonnet Black Holes in AdS Spaces

TL;DR

This work analyzes thermodynamics and phase structures of topological Gauss-Bonnet-AdS black holes with horizons of constant curvature k=0, -1, or 1. It derives exact static solutions in Einstein gravity augmented by a Gauss-Bonnet term and a negative cosmological constant, computing mass, temperature, entropy, and free energy while enforcing a ghost-free branch and the constraint $4\tilde{\alpha}/l^2 \le 1$. The results show that $k=0$ black holes have thermodynamics identical to the Einstein case with area-law entropy, and $k=-1$ black holes are always locally stable and globally preferred; for $k=1$, a novel, dimension-dependent phase structure emerges in $d=5$ with a possible locally stable small-black-hole phase under a critical coupling, while for $d\ge6$ the behavior resembles the GB-free case with a Hawking-Page transition. The findings illuminate how horizon topology and Gauss-Bonnet corrections shape holographic thermodynamics and stability in AdS spacetimes, with implications for AdS/CFT correspondence and higher-derivative gravity theories.

Abstract

We study thermodynamic properties and phase structures of topological black holes in Einstein theory with a Gauss-Bonnet term and a negative cosmological constant. The event horizon of these topological black holes can be a hypersurface with positive, zero or negative constant curvature. When the horizon is a zero curvature hypersurface, the thermodynamic properties of black holes are completely the same as those of black holes without the Gauss-Bonnet term, although the two black hole solutions are quite different. When the horizon is a negative constant curvature hypersurface, the thermodynamic properties of the Gauss-Bonnet black holes are qualitatively similar to those of black holes without the Gauss-Bonnet term. When the event horizon is a hypersurface with positive constant curvature, we find that the thermodynamic properties and phase structures of black holes drastically depend on the spacetime dimension $d$ and the coefficient of the Gauss-Bonnet term: when $d\ge 6$, the properties of black hole are also qualitatively similar to the case without the Gauss-Bonnet term, but when $d=5$, a new phase of locally stable small black hole occurs under a critical value of the Gauss-Bonnet coefficient, and beyond the critical value, the black holes are always thermodynamically stable. However, the locally stable small black hole is not globally preferred, instead a thermal anti-de Sitter space is globally preferred. We find that there 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.

Figures (9)

• Figure 1: The inverse temperature of topological black holes without the Gauss-Bonnet term. The three curves above from up to down correspond to the cases $k=-1$, $0$ and $1$, respectively.
• Figure 2: The inverse temperature of the $k=-1$ Gauss-Bonnet black holes in $d=6$ dimensions.
• Figure 3: The inverse temperature of the $k=1$ Gauss-Bonnet black holes with $\tilde{\alpha}/l^2 =0.001$. The three curves from up to down correspond to $d=5$, $6$ and $d=10$, respectively.
• Figure 4: The inverse temperature of the $k=1$ Gauss-Bonnet black holes in $d=5$ dimensions with $\tilde{\alpha}/l^2 =0.0056$.
• Figure 5: The inverse temperature of the $k=1$ Gauss-Bonnet black holes in $d=5$ dimensions. The three curves from up to down correspond to the cases with the supcritical $\tilde{\alpha}/l^2= 0.20$, critical $1/36 \approx 0.0278$, and subcritical $0.005$, respectively.