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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., Comment: 10 pages, 4 figures. Appendix added, final version

Gravitational waves are excellent tools to probe the foundations of General Relativity in the strongly dynamical and non-linear regime. One such foundation is Lorentz symmetry, which can be broken in the gravitational sector by the existence of a preferred time direction, and thus, a preferred frame at each spacetime point. This leads to a modification in the orbital decay rate of binary systems, and also in the generation and chirping of their associated gravitational waves. We here study whether waves emitted in the late, quasi-circular inspiral of non-spinning, neutron star binaries can place competitive constraints on two proxies of gravitational Lorentz-violation: Einstein-\AE{}ther theory and khronometric gravity. We model the waves in the small-coupling (or decoupling) limit and in the post-Newtonian approximation, by perturbatively solving the field equations in small deformations from General Relativity and in the small-velocity/weak-gravity approximation. We assume a gravitational wave consistent with General Relativity has been detected with second- and third-generation, ground-based detectors, and with the proposed space-based mission, DECIGO, with and without coincident electromagnetic counterparts. Without a counterpart, a detection consistent with General Relativity of neutron star binaries can only place competitive constraints on gravitational Lorentz violation when using future, third-generation or space-based instruments. On the other hand, a single counterpart is enough to place constraints that are 10 orders of magnitude more stringent than current binary pulsar bounds, even when using second-generation detectors. This is because Lorentz violation forces the group velocity of gravitational waves to be different from that of light, and this difference can be very accurately constrained with coincident observations., Comment: 21 pages, 6 figures. Submitted to Phys. Rev. D