Even as our measurements of cosmological parameters improve, the physical nature of the dark sector of the universe largely remains a mystery. Many effects of dark sector models are most prominent at very large scales and will rely on future galaxy surveys to elucidate. In this paper, we compare the topological properties of the large-scale dark matter distribution in a number of cosmological models using hydrodynamical simulations and the cosmological genus statistic. Genus curves are computed from z = 11 to 0 for λ cold dark matter (λCDM), quintessence and warm dark matter (WDM) models, over a scale range of 1-20 h-1 Mpc. The curves are analysed in terms of their Hermite spectra to describe the power contained in non-Gaussian deformations to the cosmological density field. We find that the λCDM and λWDM models produce nearly identical genus curves indicating no topological differences in structure formation. The quintessence model, which differs solely in its expansion history, produces significant differences in the strength and redshift evolution of non-Gaussian modes associated with higher cluster abundances and lower void abundances. These effects are robust to cosmic variance and are characteristically different from those produced by tweaking the parameters of a λCDM model. Given the simplicity and similarity of the models, detecting these discrepancies represents a promising avenue for understanding the effect of non-standard cosmologies on large-scale structure.