Thermodynamic and gravitational instability on hyperbolic spaces

We study the properties of anti--de Sitter black holes with a Gauss-Bonnet term for various horizon topologies (k=0, 1) and for various dimensions, with emphasis on the less well understood k=-1 solution. We find that the zero temperature (and zero energy density) extremal states are the local minima of the energy for AdS black holes with hyperbolic event horizons. The hyperbolic AdS black hole may be stable thermodynamically if the background is defined by an extremal solution and the extremal entropy is non-negative. We also investigate the gravitational stability of AdS spacetimes of dimensions D>4 against linear perturbations and find that the extremal states are still the local minima of the energy. For a spherically symmetric AdS black hole solution, the gravitational potential is positive and bounded, with or without the Gauss-Bonnet type corrections, while, when k=-1, a small Gauss-Bonnet coupling, namely, l^2 (where l is the curvature radius of AdS space), is found useful to keep the potential bounded from below, as required for stability of the extremal background.