Second law of black hole mechanics for all 2-dimensional dilaton theories

It is shown that all generalized two-dimensional dilaton theories with arbitrary matter content (with a curvature independent coupling to gravity) do not only obey a first law of black hole mechanics (which follows from Wald’s general considerations, if the entropy S is defined appropriately), but also a second law $\delta S>~0,$ provided only that the null energy condition holds and that, loosely speaking, for late times a stationary state is assumed. Also, any two-dimensional $f\left(R\right)\mathrm{}$ theory is covered. This generalizes a previous proof of Frolov to a much wider class of theories.