  We study anti-de Sitter black holes in the Einstein-Gauss-Bonnet and the
generic R^2 gravity theories, evaluate different thermodynamic quantities, and
also examine the possibilities of Hawking-Page type thermal phase transitions
in these theories. In the Einstein theory, with a possible cosmological term,
one observes a Hawking-Page phase transition only if the event horizon is a
hypersurface of positive constant curvature (k=1). But, with the Gauss-Bonnet
or/and the (Riemann)^2 interaction terms, there may occur a similar phase
transition for a horizon of negative constant curvature (k=-1). We examine the
finite coupling effects, and find that N>5 could trigger a Hawking-Page phase
transition in the latter theory. For the Gauss-Bonnet black holes, one relates
the entropy of the black hole to a variation of the geometric property of the
horizon based on first law and Noether charge. With (Riemann)^2 term, however,
we can do this only approximately, and the two results agree when, r_H>>L, the
size of the horizon is much bigger than the AdS curvature scale. We establish
some relations between bulk data associated with the AdS black hole and
boundary data defined on the horizon of the AdS geometry. Following a heuristic
approach, we estimate the difference between Hubble entropy {\cal S}_H and
Bekenstein-Hawking entropy {\cal S}_{BH} with (Riemann)^2 term, which, for k=0
and k=-1, would imply {\cal S}_{BH}\leq {\cal S}_H.
