Thermodynamic and Gravitational Instability on Hyperbolic Spaces

We study the properties of anti-de Sitter black holes for the various topologies (k=0,\pm 1) of the horizons and for various dimensions. We explore the thermodynamic and classical (in)stability of higher dimensional black holes, with emphasis on the k=-1 solution because stability of the k=(0,+1) black hole solutions is relatively more known. In particular, we show that there exists an unique k=-1 extremal black hole solution which has the lowest energy for all spacetimes in its asymptotic class. What looks encouraging is that that Gauss-Bonnet type curvature corrections to the Einstein action not only admit exact solutions but they might be crucial for stability of hyperbolic (k=-1) black holes, if we wish the latter to have. This in turn implies that the hyperbolic spacetimes can be stable thermodynamically and classically if the background is defined by an extremal solution and the extremal entropy is non-negative. For the ground state metric taken from the Einstein-Gauss-Bonnet theory, the gravitational potential can be positive and bounded from below for a small coupling \alpha (<<l^2), with $l$ being the effective curvature radius of anti-de Sitter geometry.