  We study the properties of anti--de Sitter black holes with a Gauss-Bonnet
term for various horizon topologies (k=0, \pm 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, \alpha << {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.
