In asymptotically flat space a rotating black hole cannot be in thermodynamic equilibrium because the thermal radiation would have to be co-rotating faster than light far from the black hole. However in asymptotically anti-de Sitter space such equilibrium is possible for certain ranges of the parameters. We examine the relationship between conformal field theory in rotating Einstein universes of dimensions two to four and Kerr anti-de Sitter black holes in dimensions three to five. The five dimensional solution is new. We find similar divergences in the partition function of the conformal field theory and the action of the black hole at the critical angular velocity at which the Einstein rotates at the speed of light. This should be an interesting limit in which to study large $N$ Yang-Mills.

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