  In this article, we investigate the thermodynamic topology of some black
holes, 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 generalized
off-shell Helmholtz free energy method, we calculate the thermodynamic topology
of the selected black holes in each space separately and determine their
topological classifications. We show that the addition of GB terms, dimensions,
and other factors do not affect the topological classes of black holes in both
spaces. The calculations and plots indicate that the AdS R-N and AdS EGB black
holes 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 black
holes show an interesting behavior. In addition to BBT and RPS, we also
consider the extended phase space thermodynamics (EPST) and evaluate the
thermodynamic topology for AdS EPYM black hole. The changing ($r-\tau$) in both
spaces shows similar behavior. Also, the topological number and the total
topological numbers for this black hole in the BB, RPS and EPS thermodynamics
are 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 the
Einstein-Yang-Mills black hole has thermodynamic topology equivalence in three
spaces. The present result may be due to the non-linear YM charge parameter and
the difference between the gauge and gravity corrections in the above black
holes
