  In this article, we investigate the bulk-boundary and restricted phase space
(RPS) thermodynamics of Rissner-Nordstr\"om (R-N) AdS and 6-dimensional charged
Gauss-Bonnet AdS black holes. Also, we examine the topological characteristics
of the considered black holes and compare them with the extended thermodynamics
results. In fact, we have found 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. We
also demonstrate that within the RPS formalism, there is only one critical
point with a topological charge of +1 $(Q_t=+1)$. Additionally, for 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.
