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

Abstract

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 and/or the $(\mathrm{Riemann}{)}^{2}$ interaction terms, there may occur a similar phase transition for a horizon of negative constant curvature $(k=\ensuremath{-}1).$ 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 the first law and Noether charge. With a $(\mathrm{Riemann}{)}^{2}$ term, however, we can do this only approximately, and the two results agree when ${r}_{H}\ensuremath{\gg}L,$ 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 $({\mathcal{S}}_{H})$ and Bekenstein-Hawking entropy $({\mathcal{S}}_{\mathrm{BH}})$ with a $(\mathrm{Riemann}{)}^{2}$ term, which, for $k=0$ and $k=\ensuremath{-}1,$ implies ${\mathcal{S}}_{\mathrm{BH}}<~{\mathcal{S}}_{H}.$