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 $f"(0)$ 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
$\frac{f'(T)}{4}A$.
