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This study examines the thermodynamic structure of ModMax AdS black holes, focusing on the application of non-extended entropies such as Rényi entropy (parameterized by λ ), Sharma-Mittal entropy (characterized by α and β ), and Barrow entropy (defined by δ ). Our objective is to classify the topology of these black holes and analyze how they differ from the Bekenstein-Hawking entropy. Our study reveals that the number of topological charges, correlated with the winding number, varies with the parameters of the non-extended entropies and the free parameters: pressure ( p ) , charge ( q ) , and the ModMax parameter ( γ ) . We observe that the behavior and number of topological charges change with a rise in the nonextensive parameter δ . Also, changing the parameter λ affects black hole behavior in a model transitioning from Bekenstein-Hawking entropy to Rényi entropy, and specifically, increasing λ around 0.3 changes the phase transition from first-order to second-order, as indicated by the flattening and disappearance of the τ curve. Regarding Sharma-Mittal entropy, we examine three different intervals, each showing variations in the number of topological charges and their classification. When the parameter α is greater than β , we observe two topological charges ( ω = + 1 , − 1 ) with a TTC ( W = 0 ) . When the non-extensive parameters α and β are similar in value, a distinct classification emerges, leading to a TTC of ( W = + 1 ) . Lastly, when β is greater than α , we observe one topological charge ( ω = − 1 ) with a TTC ( W = − 1 ). Bringing to light the application of non-extended entropy frameworks, such investigations into the subject of geometrothermodynamic (GT) tools governed by the Hendi, Panahiyan, Eslam Panah, and Momennia (HPEM) formalism and the sparsity of Hawking radiation are carried out. Our study provides new insights into how the application of the non-extensive entropy is interrelated within the context of black hole thermodynamics with the ModMax term by means of Duan’s ϕ techniques.
Gashti et al. (Sat,) studied this question.