Recently, the investigation of the topological classes of black holes with two different methods has attracted the attention of researchers with respect to the existence of critical points, which are important structures in the phase diagram of a thermodynamic system. One of these methods is the examination of the topological classes of black holes with the help of Duan’s topological current fmapping theory. Therefore, we consider three different types of black holes, i.e. Hayward-AdS black holes, quintessential Hayward-AdS black holes, and AdS Hayward black holes in Einstein-Gauss- Bonnet gravity. Then we explore their thermodynamic topological classes. The interesting point to note in our studies is that, unlike previous studies, all of our black holes have the same topological class, and the addition of quintessence or GBterms, etc, or dimensions does not change the topological numbers. In the end, we compare the thermodynamic topological class of our black holes with other black holes investigated in the literature
