Thermodynamics of higher-dimensional Brans-Dicke black holes in the presence of a conformal-invariant field inspired by power-Maxwell electrodynamics

By use of the conformal transformations, in addition to translating the Brans-Dicke (BD) action to the Einstein frame (EF), we introduce an electromagnetic Lagrangian which preserves conformal invariance. We solve the EF field equations, which mathematically are confronted with the problem of indeterminacy, by use of an exponential ansatz function. When the self-interacting potential is absent or is taken constant in the BD action, the exact solution of this theory is just that of Einstein-conformal-invariant theory with a trivial scalar field. This is a higher-dimensional (HD) analogue of the same considered in Ref. [R.-G. Cai, Y. S. Myung, Phys. Rev. D 56, 3466 (1997)]. The EF general solution admits two classes of black holes (BHs) with non-flat and non-AdS asymptotic behavior which can produce extreme and multi-horizon ones. We obtain the exact solutions of BD-conformal-invariant theory, by applying inverse conformal transformations, which show two classes of extreme and multi-horizon BHs too. Based on the fact that thermodynamic quantities remain unchanged under conformal transformations, we show that the first law of BH thermodynamics is valid in the Jordan frame. We analyze the thermal stability of the HD BD-conformal-invariant BHs by use of the canonical ensemble method.