The origin of entropy production in spacetime thermodynamics
  We find that the ambiguity term of approximate Killing vector field is
responsible for the entropy production term. Without the ambiguity term, pure
Einstein theory and $f(R)$ satisfy the relation of thermodynamic equilibrium.
Considering such an ambiguity term of approximate Killing vector field, we can
get the entropy production term and the entropy in $f(R)$ with a form defined
by Jacobson. In pure Einstein theory, the shear term is the only geometric
contribution of entropy production term, while in $f(R)$ it can also
contribute. We believe our approach and conclusion can be generalized to other
gravity theory.
