  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 infra-red divergences, and
all ultra-violet 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-Nordstrom-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.
