Black hole thermodynamics and massive gravity

We consider the generalized laws of thermodynamics in massive gravity. Making use of explicit black hole solutions, we devise black hole merger processes in which i) total entropy of the system decreases ii) the zero-temperature extremal black hole is created. Thus, both second and third laws of thermodynamics are violated. In both cases, the violation can be traced back to the presence of negative-mass black holes, which, in turn, is related to the violation of the null energy condition. The violation of the third law of thermodynamics implies, in particular, that a naked singularity may be created as a result of the evolution of a singularity-free state. This may signal a problem in the model, unless the creation of the negative-mass black holes from positive-mass states can be forbidden dynamically or the naked singularity may somehow be resolved in a full quantum theory.