Abstract We explore higher-dimensional black holes (BHs) using the Sharma–Mittal entropy, a widely employed measure in BH thermodynamics due to the non-extensive nature of BHs. We compute different thermodynamic quantities like temperature, heat capacity, and Gibbs free energy to study the stability of higher-dimensional BHs in Einstein–Gauss–Bonnet gravity (EGBG). We explicitly analyze the behavior for different horizon topologies (parameterized by k) in EGBG and observe that these BH exhibit both local and global instability. Additionally, we study the sparsity of Hawking radiation, noting that the sparsity parameter tends to zero as the horizon radius increases. It is also observed that emission of energy radiates most rapidly for larger BHs, which is consistent with physical observations. Furthermore, to support our analysis, we choose thermodynamical topology methods (specifically, the temperature and on-shell Gibbs free energy) to observe the stability of the aforementioned BH solutions. It is found that the total topological charge obtained from the free energy method leads to +\, 1 + 1 for specific values of k, which also gives the confirmation of the stability of these BH solutions in EGBG for a specific choice of other constant parameters.
Rani et al. (Thu,) studied this question.