In this paper, we consider a dyonic charged anti-de Sitter black hole, which is a holographic dual of a van der Waals fluid. We use logarithm-corrected entropy and study thermodynamics of the black hole and show that holographic picture is still valid. Critical behaviors and stability are also discussed. Logarithmic corrections arises due to thermal fluctuations, which are important when the size of black hole is small. So, thermal fluctuations are interpreted as a quantum effect. It means that we can see the quantum effect of a black hole, which is a gravitational system.

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 ϕ-mapping 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 GB terms, 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.