We study anti-de Sitter black holes and evaluate different thermodynamic quantities in the EinsteinGauss-Bonnet and the general R 2 gravity theories. We examine the possibility of Hawking-Page type thermal phase transitions between AdS black hole and thermal anti-de Sitter space in such theories. In 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 in EinsteinGauss-Bonnet gravity there can occur a similar transition even for a horizon of negative constant curvature (k = −1), which may allow one to study the boundary conformal theory with different background geometries. For the Gauss-Bonnet black holes, one can relate the entropy of the black hole as measured at horizon to a variation of the geometric property of the horizon based on first law and Noether charge. With (Riemann) 2 terms, however, we can do this only approximately, and the two results agree in the limit r H >> L, the size of the horizon is much bigger than the AdS curvature. In (Riemann) 2 gravity, we establish certain relations between bulk data associated with the AdS black hole in five dimensions and boundary data defined on the horizon of the AdS geometry , in which case we do not expect a sensible holographic dual. We also give a heuristic approach to estimate the difference between Hubble entropy and Bakenstein-Hawking entropy with (Riemann) 2 term.