Thermodynamics of Charged Brans-Dicke AdS Black Holes

It is well-known that in four dimensions, black hole solution of the Brans-Dikce-Maxwell equations is just the Reissner-Nordstrom solution with a constant scalar field. However, in $n\geq4$ dimensions, the solution is not yet the $(n+1)$-dimensional Reissner-Nordstrom solution and the scalar field is not a constant in general. In this paper, by applying a conformal transformation to the dilaton gravity theory, we derive a class of black hole solutions in $(n+1)$-dimensional $(n\geq 4)$ Brans-Dikce-Maxwell theory in the background of anti-de Sitter universe. We obtain the conserved and thermodynamic quantities through the use of the Euclidean action method. We find a Smarr-type formula and perform a stability analysis in the canonical ensemble. We find that the solution is thermally stable for small $\alpha$, while for large $\alpha$ the system has an unstable phase, where $\alpha $ is a coupling constant between the scalar and matter field.