  We study the thermodynamic and gravitational stability of Kerr anti-de Sitter
black holes in five and higher dimensions. We show, in the case of equal
rotation parameters, $a_i=a$, that the Kerr-AdS background metrics become
stable, both thermodynamically and gravitationally, when the rotation
parameters $a_i$ take values comparable to the AdS curvature radius. In turn, a
Kerr-AdS black hole can be in thermal equilibrium with the thermal radiation
around it only when the rotation parameters become not significantly smaller
than the AdS curvature radius. We also find with equal rotation parameters that
a Kerr-AdS black hole is thermodynamically favored against the existence of a
thermal AdS space, while the opposite behavior is observed in the case of a
single non-zero rotation parameter. The five dimensional case is however
different and also special in that there is no high temperature thermal AdS
phase regardless of the choice of rotation parameters. We also verify that at
fixed entropy, the temperature of a rotating black hole is always bounded above
by that of a non-rotating black hole, in four and five dimensions, but not in
six and more dimensions (especially, when the entropy approaches zero or the
minimum of entropy does not correspond to the minimum of temperature). In this
last context, the six dimensional case is marginal.
