Thermodynamic topology of black hole in Massive gravity with generalized Barrow modification of entropy

The goal of our work is to explore the topology classifications of black holes in Massive gravity. We use Duan's ϕ-mapping theory to study the topology classes. This tool helps us to compute the integer numbers to characterize the topological features of the black hole. Our analysis is carried out across two different thermodynamic ensembles. These statistical ensembles are the canonical ensemble and the mixed canonical ensemble. Every ensemble corresponds to a different thermodynamic condition, allowing us to investigate how the black holes respond to other physical conditions. Thus, employing these analytical techniques, it is expected that further insights into topological and thermodynamic aspects of black holes in these modified gravity frameworks can be explored. This work helps to shed light on black hole dependencies on gravitational theories and conditions in thermal processes and thus expands the knowledge, proposing a new essence of black holes.