We study the thermodynamics of four-dimensional Kerr-Newman-AdS black holes
both in the canonical and the grand-canonical ensemble. The stability cond
itions are investigated, and the complete phase diagrams are obtained, whic
h include the Hawking-Page phase transition in the grand-canonical ensemble
. In the canonical case, one has a first-order transition between small and
large black holes, which disappears for a sufficiently large electric char
ge or angular momentum. This disappearance corresponds to a critical point
in the phase diagram. Via the AdS/CFT conjecture, the obtained phase struct
ure is also relevant for the corresponding conformal held theory living in
a rotating Einstein universe, in the presence of a global background U(1) c
urrent. An interesting limit arises when the black holes presence some supe
rsymmetry. These BPS black holes correspond to highly degenerate zero-tempe
rature stares in the dual CFT, which lives in an Einstein universe rotating
with the speed of light.