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Under broad assumptions breaking of Lorentz invariance in gravitationaltheories leads to tension with unitarity because it allows for processes thatapparently violate the second law of thermodynamics. The crucial ingredient ofthis argument is the existence of black hole solutions with the interiorshielded from infinity by a causal horizon. We study how the paradox can beresolved in the healthy extension of Horava gravity. To this aim we analyzeclassical solutions describing large black holes in this theory with theemphasis on their causal structure. The notion of causality is subtle in thistheory due to the presence of instantaneous interactions. Despite this fact, wefind that within exact spherical symmetry a black hole solution contains aspace-time region causally disconnected from infinity by a surface of finitearea -- the `universal horizon'. We then consider small perturbations ofarbitrary angular dependence in the black hole background. We argue thataspherical perturbations destabilize the universal horizon and, at non-linearlevel, turn it into a finite-area singularity. The causal structure of theregion outside the singularity is trivial. If the higher-derivative terms inthe equations of motion smear the singularity while preserving the trivialcausal structure of the solutions, the thermodynamics paradox would beobviated. As a byproduct of our analysis we prove that the black holes do nothave any non-standard long-range hair. We also comment on the relation withEinstein-aether theory, where the solutions with universal horizon appear to bestable.

Horava gravity vs. thermodynamics: the black hole case