  We compute the renormalized vacuum polarization for a massless, conformally
coupled scalar field on asymptotically anti-de Sitter black hole backgrounds.
Mixed (Robin) boundary conditions are applied on the spacetime boundary. We
consider black holes with nonspherical event horizon topology as well as
spherical event horizons. The quantum scalar field is in the Hartle-Hawking
state, and we employ Euclidean methods to calculate the renormalized
expectation values. Far from the black hole, we find that the vacuum
polarization approaches a finite limit, which is the same for all boundary
conditions except Dirichlet boundary conditions.
