  We investigate thermodynamics of static and spherically symmetric black holes
(BHs) in the Horndeski theories. Because of the presence of the
higher-derivative interactions and the nonminimal derivative couplings of the
scalar field, the standard Wald entropy formula may not be directly applicable.
Hence, following the original formulation by Iyer and Wald, we obtain the
differentials of the BH entropy and the total mass of the system in the
Horndeski theories, which lead to the first-law of thermodynamics via the
conservation of the Hamiltonian. Our formulation covers the case of the static
and spherically symmetric BH solutions with the static scalar field and those
with the linearly time-dependent scalar field in the shift-symmetric Horndeski
theories. We then apply our results to explicit BH solutions in the Horndeski
theories. In the case of the conventional scalar-tensor theories and the
Einstein-scalar-Gauss-Bonnet theories, we recover the BH entropy obtained by
the Wald entropy formula. In the shift-symmetric theories, in the case of the
BH solutions with the static scalar field we show that the BH entropy follows
the ordinary area law even in the presence of the nontrivial profile of the
scalar field. On the other hand, in the case of the BH solutions where the
scalar field linearly depends on time, i.e., the stealth Schwarzschild and
Schwarzschild-(anti-) de Sitter solutions, the BH entropy also depends on the
profile of the scalar field. By use of the entropy, we find that there exists
some range of the parameters in which Schwarzschild$-$(AdS) BH with non-trivial
scalar field is thermodynamically stable than Schwarzschild$-$(AdS) BH without
scalar field in general relativity.
