High-order QED correction impacts on phase transition of the Euler-Heisenberg dS spacetime

Recent studies have demonstrated that AdS black holes possess the basic characteristics of a standard thermodynamic system. Concurrently, the thermodynamic properties of spacetimes featuring multiple horizons with distinct radiation temperatures have also attracted research interest. In this work, considering the high-order quantum electrodynamics (QED) correction, we initially establish an equivalent thermodynamic system for the coexistence region of black hole and cosmological horizons. On this basis, we conduct a detailed investigation into the thermodynamic properties of this dual-horizon coexistence region. Our results demonstrate that this equivalent thermodynamic system exhibits van der Waals-like thermodynamic behavior. Furthermore, we introduce a nonlinear parameter $\gamma$ to analyze its impact on phase transitions within the equivalent thermodynamic system. Under specific conditions, the system undergoes first- or second-order phase transitions for $\gamma=0$, and zeroth- or second-order phase transitions for $\gamma\neq0$. Finally, by utilizing the generalized off-shell Helmholtz free energy within the thermodynamic topological framework for black holes, we extend this methodology to investigate the topological properties of de Sitter (dS) spacetime. We compute the topological number characterizing the coexistence region of dual horizons in Euler-Heisenberg (EH) dS spacetime using equivalent thermodynamic state parameters. Additionally, we investigate the influence of the nonlinear parameter $\gamma$ on the thermodynamic characteristics of the equivalent system.