Thermodynamics of Euler–Heisenberg black holes in the Einstein-dilaton gravity theory

In this paper, we investigate new dilaton black holes (BHs) in the presence of nonlinear Euler–Heisenberg (EH) electrodynamics and study the related thermodynamics. To achieve this, we solve the field equations in a static, spherically symmetric spacetime and obtain a set of EH BH solutions that behave unusually at infinity. By using the plots, we show that our solutions are capable of indicating multi-horizon BHs, which reflect the quantum anti-evaporation phenomena. We calculate thermodynamic parameters such as charge, electric potential, mass, entropy, and temperature by using the appropriate methods. We show that the first law of thermodynamics (FLT) is valid provided that the scalar-electromagnetic coupling parameter [Formula: see text] is related to the dilaton parameter [Formula: see text] via the relation [Formula: see text]. The mathematical interpretation of this relation is that the EH nonlinearity parameter is not a conformal-invariant quantity. Based on this result, we construct the physically acceptable exact solutions of this theory. Then, we investigate the thermal stability of charged Einstein-dilaton BHs, by using the canonical ensemble method.