Violation of the first law of black hole thermodynamics in f(T) gravity

We prove that, in general the first law of black hole thermodynamnics, delta Q = T delta S, is violated in f(T) gravity. As a result, it is possible that there exists entropy production, which implies that the black hole thermodynamics can be in non-equilibrium even in the static spacetime. This feature is very different from that of f(R) or that of other higher derivative gravity theories. We find that the violation of first law results from the lack of local Lorentz invariance in f(T) gravity. By investigating two examples, we note that f ''(0) should be negative in order to avoid the naked singularities and superluminal motion of light. When f ''(T) is small, the entropy of black holes in f(T) gravity is approximatively equal to (f'(T))/(4)A