  In this paper, we use the generalized off-shell Helmholtz free energy method
to explore the thermodynamic properties of Bardeen black holes (BD BHs) from a
topological perspective based on Duan's topological current $\phi$-mapping. We
consider various structures of BD BHs, including regular BD-AdS BHs, BD-AdS BHs
in Kiselev's model of quintessence, BD BHs in massive gravity (MG), and BD BHs
in 4D Einstein-Gauss-Bonnet (EGB) gravity. We demonstrate that these BHs have
one topological classification (TC), i.e., TC is +1 for all BHs considered, and
the addition of MG or GB terms, etc., does not change the topological numbers.
