  In this study, we employ the thermodynamic topological method to classify
critical points for the dyonic AdS black holes with quasitopological
electromagnetism in the Einstein-Gauss-Bonnet background. To this end, we find
a small/large black hole phase transition in all dimensions of space-time, the
existence of a conventional critical point implies a total topological charge
of $Q_t=-1$. The coupling constant $\alpha$ gives rise to a more intricate
phase structure, with the emergence of a triple points requires $\alpha\geq0.5$
and $d=6$. Interestingly, the condition for the occurrence of
small/intermediate/large phase transition is that the coupling constant a takes
a special value ($\alpha=0.5$), the two conventional critical points
$CP_{1},CP_{2}$ of the black hole are physical critical point, and the novel
critical point $CP_{3}$ that lacks the capability to minimize the Gibbs free
energy. The critical points $CP_{1}$ and $CP_{2}$ are observed to occur at the
maximum extreme points of temperature in the isobaric curve, while the critical
point $CP_{3}$, emerges at the minimum extreme points of temperature.
Furthermore, the number of phases at the novel critical point exhibits an
upward trend, followed by a subsequent decline at the conventional critical
points. With the increase of the coupling constant ($\alpha = 1$), although the
system has three critical points, only the conventional $CP_{1}$ is a
(physical) critical point, and the conventional $CP_{2}$ serves as the phase
annihilation point. This means that the coupling constant $\alpha$ has a
non-negligible effect on the phase structure.
