The thermodynamic potentials of Kerr-AdS black holes and their CFT duals

String or M-theory in the background of Kerr-AdS black holes is thought to be dual to the large N limit of certain conformal field theories on a rotat ing sphere at finite temperature. The five dimensional black hole is associ ated to N = 4 supersymmetric Yang-Mills theory on a rotating three-sphere a nd the four dimensional one to the superconformal field theory of coincidin g M2 branes on a rotating two-sphere. The thermodynamic potentials can be e xpanded in inverse powers of the radius of the sphere. We compute the leadi ng and subleading terms of this expansion in the field theory at one loop a nd compare them to the corresponding supergravity expressions. The ratios b etween these terms at weak and strong coupling turn out not to depend on th e rotation parameters in the case of N = 4 SYM. For the field theory living on one M2 brane we find a subleading logarithmic term. No such term arises from the supergravity calculation.