Thermodynamic topology and phase space analysis of AdS black holes through non-extensive entropy perspectives

In this paper, we study the thermodynamic topology of AdS Einstein–power–Yang–Mills black holes, examining them through both the bulk-boundary and restricted phase space (RPS) frameworks. We consider various non-extensive entropy models, including Barrow ( ), Rényi ( ), Sharma–Mittal ( , ), Kaniadakis ( ), and Tsallis-Cirto entropy ( ). Initially, we analyze the thermodynamic topology within the bulk-boundary framework. Our findings highlight the influence of free parameters on topological charges. We observe two topological charges with respect to the non-extensive Barrow parameter and also with ( ) in Bekenstein–Hawking entropy. For Rényi entropy, different topological charges are observed depending on the value of the with a notable transition from three topological charges to a single topological charge as increases. Also, by setting to zero results in two topological charges . Sharma–Mittal entropy exhibits three distinct ranges of topological charges influenced by the and with different classifications viz, if exceeds , we will have ; if , we have ; and if exceeds , we obtain . Also, Kaniadakis entropy shows variations in topological charges; viz., we observe for any acceptable value of K, except when , where a single topological charge appears. In the case of Tsallis-Cirto entropy, for small parameter values, we have and when increases to 0.9, we will have . A particularly intriguing aspect of this research is its application to the RPS framework. When we extend our analysis to this space using the specified entropies, we find that the topological charge consistently remains independent of the specific values of the free parameters for Rényi, Sharma–Mittal, and Tsallis–Cirto. Additionally, for Barrow entropy in RPS, when increases from 0 to 0.8, the number of topological charges rises. Finally for Kaniadakis entropy, at small values of , we observe . However, as the non-extensive parameter increases, we encounter different topological charges and classifications with .