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

We study anti-de Sitter black holes in the Einstein-Gauss-Bonnet and the generic $R^2$ gravity theories, and evaluate different thermodynamic quantities and also examine the possibilities of Hawking-Page type thermal phase transitions between AdS black hole and thermal anti-de Sitter space 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, in the Einstein-Gauss-Bonnet theory there may occur a similar phase transition even for a horizon of negative constant curvature ($k=-1$), which allows 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 $({Riemann})^2$ 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 that the AdS curvature scale. With $({Riemann})^2$ term, 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. 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 $k=-1$, implies ${\cal S}_{BH}\leq {\cal S}_H$.