In this study, we investigate the bulk-boundary and restricted phase space (RPS) thermodynamics of Rissner-Nordström (R-N) AdS and 6-dimensional charged Gauss-Bonnet AdS black holes. Additionally, we examine the topological characteristics of the considered black holes and compare them with the results of extended thermodynamics. We determine that the topological behavior of the bulk-boundary thermodynamics is the same as that of the extended thermodynamics, whereas the RPS thermodynamics exhibits a distinct behavior. Furthermore, we demonstrate that within the RPS formalism, there is only one critical point with a topological charge of +1 . Moreover, in the RPS formalism, the inclusion of higher-derivative curvature terms in the form of Gauss- Bonnet gravity does not alter the topological classification of critical points in charged AdS black holes
