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Vacuum energy in Kerr-AdS black holes

We compute the vacuum energy for Kerr black holes with anti-de Sitter (AdS) asymptotics in dimensions $5\leq D\leq 9$ 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 global AdS spacetime written in oblate spheroidal coordinates. We find that the zero-point energy for higher-dimensional Kerr-AdS reduces to the one of Schwarzschild-AdS black hole when all the rotation parameters are equal to each other, 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.