Black hole thermodynamics in the Brans–Dicke–Maxwell theory

It has been shown that the exact solutions of four-dimensional (4D) Brans–Dicke–Maxwell (BDM) theory is nothing other than Reissner–Nordström (RN) black hole (BH)s coupled to a trivial constant scalar field (Cai and Myung in Phys Rev D 56:3466, 1997). Here, we show that it is the case only when the scalar potential is taken constant or equal to zero. Then, through obtaining the exact solutions, in the presence of a scalar potential, we show that this theory admits two classes of novel BH solutions which have been affected by a nontrivial scalar hair. Due to conformal invariance of Maxwell’s electrodynamics, multi-horizon BHs can occur which implies the anti-evaporation quantum effect. Inclusion of the scalar hair makes the asymptotic behavior of the solutions non-flat and non-AdS. Our novel solutions recover the RN-AdS BHs when the scalar field is turned off. Thermodynamic quantities of the 4D BDM BHs have been calculated by use of the appropriate methods and under the influence of scalar field. Then, by use of a Smarr-type mass formula, it has been found that the first law of BH thermodynamics is valid for our novel BHs. Thermal stability of the BDM BHs has been analyzed by use of the canonical ensemble and geometrical methods, comparatively.