  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.
