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In this article, we investigate the thermodynamic topology of some blackholes, namely AdS Reissner Nordstrom (R-N), AdS Einstein-Gauss-Bonnet (EGB),and AdS Einstein-power-Yang-Mills (EPYM), from different frameworks:bulk-boundary (BB) and restricted phase space (RPS). Using the generalizedoff-shell Helmholtz free energy method, we calculate the thermodynamic topologyof the selected black holes in each space separately and determine theirtopological classifications. We show that the addition of GB terms, dimensions,and other factors do not affect the topological classes of black holes in bothspaces. The calculations and plots indicate that the AdS R-N and AdS EGB blackholes show similar behavior and their topological numbers sets in both spaces,i.e., BB and RPS, are similar and equal to ($W=+1$). However, AdS EPYM blackholes show an interesting behavior. In addition to BBT and RPS, we alsoconsider the extended phase space thermodynamics (EPST) and evaluate thethermodynamic topology for AdS EPYM black hole. The changing ($r-\tau$) in bothspaces shows similar behavior. Also, the topological number and the totaltopological numbers for this black hole in the BB, RPS and EPS thermodynamicsare completely same, i,e., $(\omega_{BBT}=\omega_{RPS}=\omega_{EPST}=+1, -1)$or $W_{BBT}=W_{RPS}=W_{EPST}=0$. An important point is that theEinstein-Yang-Mills black hole has thermodynamic topology equivalence in threespaces. The present result may be due to the non-linear YM charge parameter andthe difference between the gauge and gravity corrections in the above blackholes

Thermodynamic topology of black holes from bulk-boundary, extended, and restricted phase space perspectives