Black hole thermodynamics in generalized Proca theories

We investigate thermodynamics of static and spherically symmetric black holes (BHs) in generalized Proca (GP) theories by applying the Iyer-Wald prescription. BH solutions in GP theories are divided into the two classes. The first class corresponds to the solutions obtained by the direct promotion of the BH solutions in shift-symmetric Horndeski theories, while the second class consists of the solutions which are obtained only in GP theories and contain a nonzero electromagnetic field. For BH solutions in the first class, we confirm that the BH entropy and its thermodynamic mass remain the same as those in the counterpart solution in shift-symmetric Horndeski theories. We also calculate the thermodynamical variables of the several static and spherically symmetric BH solutions in the second class and investigate the thermodynamical stability when there exist two BH solutions.