  Under broad assumptions breaking of Lorentz invariance in gravitational
theories leads to tension with unitarity because it allows for processes that
apparently violate the second law of thermodynamics. The crucial ingredient of
this argument is the existence of black hole solutions with the interior
shielded from infinity by a causal horizon. We study how the paradox can be
resolved in the healthy extension of Horava gravity. To this aim we analyze
classical solutions describing large black holes in this theory with the
emphasis on their causal structure. The notion of causality is subtle in this
theory due to the presence of instantaneous interactions. Despite this fact, we
find that within exact spherical symmetry a black hole solution contains a
space-time region causally disconnected from infinity by a surface of finite
area -- the `universal horizon'. We then consider small perturbations of
arbitrary angular dependence in the black hole background. We argue that
aspherical perturbations destabilize the universal horizon and, at non-linear
level, turn it into a finite-area singularity. The causal structure of the
region outside the singularity is trivial. If the higher-derivative terms in
the equations of motion smear the singularity while preserving the trivial
causal structure of the solutions, the thermodynamics paradox would be
obviated. As a byproduct of our analysis we prove that the black holes do not
have any non-standard long-range hair. We also comment on the relation with
Einstein-aether theory, where the solutions with universal horizon appear to be
stable.
