Anti-de Sitter Black Holes, Thermal Phase Transition and Holography in Higher Curvature Gravity

We study anti-de Sitter black holes and evaluate different thermodynamic quantities in the Einstein-Gauss-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 Einstein-Gauss-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.