Does the generalized second law hold in the form of time derivative expression?

We investigate whether the generalized second law is valid in the case of two-dimensional black hole spacetimes, irrespective of models. A time derivative form of the generalized second law is formulated and its validity is argued in two settings: the accessible region has (i) a finite spatial region and (ii) an infinite one. In case (i), it is pointed out that the violation of the law might occur. In case (ii), which corresponds to the standard setting in arguing the generalized second law, it is shown that the law holds for well-known vacuum states. However, even in this case, it seems to be violated for some class of states. A method to resolve this difficulty is also presented and discussed.