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We study anti-de Sitter black holes in the Einstein-Gauss-Bonnet and thegeneric R^2 gravity theories, evaluate different thermodynamic quantities, andalso examine the possibilities of Hawking-Page type thermal phase transitionsin these theories. In the Einstein theory, with a possible cosmological term,one observes a Hawking-Page phase transition only if the event horizon is ahypersurface of positive constant curvature (k=1). But, with the Gauss-Bonnetor/and the (Riemann)^2 interaction terms, there may occur a similar phasetransition for a horizon of negative constant curvature (k=-1). We examine thefinite coupling effects, and find that N>5 could trigger a Hawking-Page phasetransition in the latter theory. For the Gauss-Bonnet black holes, one relatesthe entropy of the black hole to a variation of the geometric property of thehorizon 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, thesize of the horizon is much bigger than the AdS curvature scale. We establishsome relations between bulk data associated with the AdS black hole andboundary data defined on the horizon of the AdS geometry. Following a heuristicapproach, we estimate the difference between Hubble entropy {\cal S}_H andBekenstein-Hawking entropy {\cal S}_{BH} with (Riemann)^2 term, which, for k=0and k=-1, would imply {\cal S}_{BH}\leq {\cal S}_H.Comment: 22 pages, Revtex 4, 12+1 figures, references added, section V  extended. To appear in PR

Anti–de Sitter black holes, thermal phase transition, and holography in higher curvature gravity