Thermodynamic topology of black holes in F(R)-Euler–Heisenberg gravity’s Rainbow

The topology of black hole thermodynamics is a fascinating area of study that explores the connections between thermodynamic properties and topological features of black holes. It often involves analyzing critical points in the phase diagrams of black holes and assigning topological charges to these points. One significant approach is based on Duan’s topological current [Formula: see text]-mapping theory, which introduces the concept of topological charges to critical points in black hole thermodynamics. We successfully derive the field equations for [Formula: see text]-Euler–Heisenberg theory, providing a framework for studying the interplay between modified gravity and nonlinear electromagnetic effects. We obtain an analytical solution for a static, spherically symmetric, energy-dependent black hole with constant scalar curvature. Also, our analysis of black holes in F(R)-Euler–Heisenberg gravity’s Rainbow reveals significant insights into their topological properties. We identified the total topological charges by examining the normalized field lines along various free parameters. Our findings indicate that the parameters [Formula: see text] and [Formula: see text] influence the topological charges. These results are comprehensively summarized in Tables. In examining the photon sphere within this model, the sign of the parameter [Formula: see text] plays a crucial role in determining whether the model adopts a dS or AdS configuration. An interesting characteristic of this model is that, in its AdS form, it avoids the formation of naked singularity regions, which sets it apart from many other models. Typically, varying parameter values in other models can result in the division of space into regions of black holes and naked singularities. However, this model consistently retains its black hole behavior by featuring an unstable photon sphere, regardless of parameter values within the acceptable range. In its dS form, the behavior of the model’s photon sphere remains consistent with other dS models and does not exhibit unique differences.