On the first law of black holes with a universal horizon

In Lorentz violating theories of gravitation with a preferred foliation a notion of black hole is still possible, despite the presence of infinitely fast propagating modes. Such event horizons are known as universal horizons. Their discovery poses the question of whether they satisfy mechanical laws, analogous to the ones of Killing horizons in Lorentz symmetric theories, and whether they admit a thermodynamical interpretation. In this paper we study the viability of the first law for several exact universal horizon solutions previously derived in the literature. Our results show that a simple mechanical and thermodynamical interpretation is problematic in these cases, and call for a more systematic study of rotating universal horizons.