Black hole thermodynamics in the four-dimensional Brans–Dicke–Euler–Heisenberg theory

Abstract In this paper, we investigate the thermodynamics of charged Brans–Dicke (BD) black holes (BHs) in the presence of Euler-Heisenberg (EH) electrodynamics. We write the action and discuss that the equations of electromagnetic, scalar, and gravitational fields are strongly coupled, which make them difficult to be solved directly. To remove this problem, we use the conformal transformations (CTs) to transfer Jordan frame (JF) action to that of Einstein frame (EF). We solve the equations and introduce two new classes of BHs in the Einstein-dilaton theory. By detailed calculations, we show that if the EH nonlinearity parameter is treated as a conformal-invariant quantity, the first law of thermodynamics (FLT) is violated. To solve this problem, we allow the nonlinear parameter to change under CTs. Then, by using this proposal, we obtain the corrected solutions by exactly solving the field equations. Our solutions are asymptotically unusual and capable of exhibiting extreme and multi-horizon dilaton BHs. Then, we calculate the thermodynamic quantities and show that the FLT holds for the new EH BHs. The thermal stability or phase transition of the BHs is analyzed by using the canonical ensemble method. Finally, we introduce new EH-BD BHs solutions, from their EF counterparts, by reversing the transformations. We explore thermodynamic quantities, FLT and thermal stability of the BD BHs by utilizing the appropriate methods.