Five-dimensional Kerr-AdS black holes, first law and counterterms

Using counterterm subtraction technique we calculate the action and other thermodynamical quantities for the general Kerr-AdS black hole in five dimensions. We show that the resulting thermodynamical quantities do satisfy the first law of thermodynamics upon choosing the non-rotating Einstein space as a boundary. Furthermore, the vacuum energies and conformal anomalies calculated from both sides of the AdS/CFT duality match exactly. Comparing this result to previous known results, one is lead to raise the question: why some choices of the boundary metric produce quantities that do not satisfy the first law? Answering such a question will help us understand the relevance of the AdS vacuum energy on the gravity side.