Four-dimensional scalar–tensor black holes charged with a corrected Maxwell filed

We study thermodynamics and thermal stability of four-dimensional scalar–tensor (ST) black holes (BHs) under the influence of first-order corrected Maxwell’s electrodynamics. The related field equations are too complex for analytical solutions and conformal transformations (CT) is a useful mathematical tool for translating the ST action from Jordan to the Einstein frame. We solve the field equations in the Einstein frame and introduce two classes of BHs in the Einstein-dilaton theory with unusual asymptotic. After calculating thermodynamic quantities and confirming validity of the thermodynamical first law (TFL), we perform a thermal stability analysis for the Einstein-dilaton BHs by using the canonical ensemble method. Then, we introduce the ST BHs by using the inversed CT. Direct calculations show that thermodynamic quantities remain unchanged and TFL is valid for the ST BHs. Also, thermal stability of the ST BHs is just like that of Einstein-dilaton ones.