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-AEther 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 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.

By regarding the Einstein equations as equation(s) of state, we demonstrate that a full cohomogeneity horizon first law can be derived in horizon thermodynamics. In this approach both the entropy and the free energy are derived concepts, while the standard (degenerate) horizon first law is recovered by a Legendre projection from the more general one we derive. These results readily generalize to higher curvature gravities and establish a way of how to formulate consistent black hole thermodynamics without conserved charges.

The Newman-Janis algorithm has been widely used to construct rotating black hole solutions from non-rotating counterparts. While this algorithm was developed within General Relativity, it has more recently been applied to non-rotating solutions in modified gravity theories. We find that the application of the Newman-Janis algorithm to an arbitrary non-GR spherically-symmetric solution introduces pathologies in the resulting axially-symmetric metric. This then establishes that, in general, the Newman-Janis algorithm should not used to construct rotating black hole solutions outside of General Relativity.

We study P −V criticality of black holes in Lovelock gravities in the context of horizon thermodynamics. The corresponding first law of horizon thermodynamics emerges as one of the Einstein-Lovelock equations and assumes the universal (independent of matter content) form δE = T δS − P δV , where P is identified with the total pressure of all matter in the spacetime (including a cosmological constant Λ if present). We compare this approach to recent advances in extended phase space thermodynamics of asymptotically AdS black holes where the 'standard' first law of black hole thermodynamics is extended to include a pressure-volume term, where the pressure is entirely due to the (variable) cosmological constant. We show that both approaches are quite different in interpretation. Provided there is sufficient non-linearity in the gravitational sector, we find that horizon thermodynamics admits the same interesting black hole phase behaviour seen in the extended case, such as a Hawking-Page transition, Van der Waals like behaviour, and the presence of a triple point. We also formulate the Smarr formula in horizon thermodynamics and discuss the interpretation of the quantity E appearing in the horizon first law.