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

We prove that, in general, the first law of black hole thermodynamics, $\delta Q=T\delta S$, 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 $f\mathrm{"}(0)$ should be negative in order to avoid the naked singularities and superluminal motion of light. When $f\mathrm{"}(T)$ is small, the entropy of black holes in $f(T)$ gravity is approximatively equal to $\frac{f^{\mathrm{\prime }}(T)}{4}A$.