  We study a modified horizon thermodynamics and the associated criticality for
rotating black hole spacetimes. Namely, we show that under a virtual
displacement of the black hole horizon accompanied by an independent variation
of the rotation parameter, the radial Einstein equation takes a form of a
"cohomogeneity two" horizon first law, $dE=TdS+\Omega dJ-\sigma dA$, where $E$
and $J$ are the horizon energy (an analogue of the Misner-Sharp mass) and the
horizon angular momentum, $\Omega$ is the horizon angular velocity, $A$ is the
horizon area, and $\sigma$ is the surface tension induced by the matter fields.
For fixed angular momentum, the above equation simplifies and the more familiar
(cohomogeneity one) horizon first law $dE=TdS-PdV$ is obtained, where $P$ is
the pressure of matter fields and $V$ is the horizon volume. A universal
equation of state is obtained in each case and the corresponding critical
behavior is studied.
