Renormalized black hole entropy in anti-de Sitter space via the ``brick wall'' method

We consider the entropy of a quantum scalar field on a background black hole geometry in asymptotically anti-de Sitter space-time, using the ``brick wall'' approach. In anti-de Sitter space, the theory has no infrared divergences, and all ultraviolet divergences can be absorbed into a renormalization of the coupling constants in the one-loop effective gravitational Lagrangian. We then calculate the finite renormalized entropy for the Schwarzschild-anti-de Sitter and extremal Reissner-Nordström-anti-de Sitter black holes, and show that, at least for large black holes, the entropy is entirely accounted for by the one-loop Lagrangian, apart possibly from terms proportional to the logarithm of the event horizon radius. For small black holes, there are indications that non-perturbative quantum gravity effects become important.