Vacuum energy in Kerr-AdS black holes

We compute the vacuum energy for Kerr black holes with anti-de Sitter (AdS) asymptotics in dimensions 5 <= D <= 9 with all rotation parameters. The calculations are carried out employing an alternative regularization scheme for asymptotically AdS gravity, which considers supplementing the bulk action with counterterms which are a given polynomial in the extrinsic and intrinsic curvatures of the boundary (also known as Kounterterms). The Kerr-Schild form of the rotating solutions enables us to identify the vacuum energy as coming from the part of the metric that corresponds to a global AdS spacetime written in oblate spheroidal coordinates. We find that the zero-point energy for higher-dimensional Kerr-AdS reduces to one of a Schwarzschild-AdS black hole when all the rotation parameters are equal to each other, a fact that is well known in five dimensions. We also sketch a compact expression for the vacuum energy formula in terms of asymptotic quantities that might be useful to extend the computations to higher odd dimensions.