Thermodynamics of scalar-tensor-Maxwell black holes

The action of the charged Einstein-dilaton gravity theory is obtained from that of the scalar-tensor gravity theory by use of the transformations defined as $g_{\mu \nu }\to {\Omega }^2{g}_{\mu \nu }$ and $A_{\mu }\to A_{\mu }$. It is shown that, under these conformal transformations, the Lagrangian of Maxwell's electrodynamics remains invariant in four-dimensional space-times. The related field equations are solved in the framework of Einstein-dilaton theory, and the dilatonic potential is obtained as a linear combination of three Liouville-type potentials. Two classes of novel charged black holes are identified as the exact solution to the field equations of the Einstein-Maxwell-dilaton theory. We calculate the conserved and thermodynamic quantities of dilaton black holes and show that the first law of black hole thermodynamics is valid in its standard form. Also, the thermodynamic stability or the phase transition of dilaton black holes is analyzed by use of the canonical ensemble method. Then, making use of the inverse conformal transformations, two classes of charged scalar-tensor black holes are obtained from their Einstein-dilaton counterparts and their thermodynamic properties as well as thermodynamic phase transitions are investigated.