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The Jordan frame (JF) field equations of scalar-tensor (ST) theory are strongly coupled and, the exact solutions cannot be obtained easily. By using the conformal transformation (CT), the ST action has been translated to the Einstein frame (EF) where the theory is known as the Einstein-dilaton (Ed) gravity. Also, an Formula: see text-dimensional electromagnetic Lagrangian has been introduced which remains invariant under CT. The Ed-conformal-invariant field equations, which are confronted with the mathematical indeterminacy problem, have been solved by use of a power-law ansatz function. We have introduced two classes of black holes (BHs) which are asymptotically non-flat and non-AdS. The Ed exact solutions can produce BHs with three, two, one and without horizons. By calculating the thermodynamic quantities, and making use of the Smarr mass relation it has been shown that the thermodynamical first law is valid in the EF. Thermal stability of Ed BHs has been analyzed by considering specific heats, thermodynamic Ricci scalars and Gibbs free energies, separately. Then using the inverse CTs, the ST exact solutions have been obtained which show two classes of horizonless, one-horizon, two-horizon and three-horizon BHs. We found that CTs preserve thermodynamic quantities and, thermodynamic properties of the ST BHs are just like those of Ed ones.
M. Dehghani (Wed,) studied this question.