Thermodynamic analysis of Magnetically charged Euler–Heisenberg black holes with scalar hair via Renyi-entropy using logarithmic correction

In this article, we investigate the thermodynamic properties of the Euler–Heisenberg black hole solution with magnetic charge and scalar hair. We also examine the evaluation of thermal fluctuations, Gibbs free energy, and energy emission. The Hawking temperature, geometric mass, and heat capacity are among the numerous factors that are computed to evaluate local and global thermodynamic stability. The first law of thermodynamics is applied to determine the temperature of the black hole, and the energy emission rate is also calculated. We investigate the phase transition behavior of the Euler–Heisenberg black hole with scalar hair by computing the Gibbs free energy, with particular attention to characteristics like eating tails. We also obtain the corrected entropy to investigate the impact of thermal fluctuations on massive and small black holes. Notably, we contrast the outcomes for large and small black holes in order to examine the effects of correction terms on the thermodynamic system.