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We explore the assumption, widely used in many astrophysical calculations, that the stellar initial mass function (IMF) is universal across all galaxies. By considering both a canonical broken-power-law IMF and a non-universal IMF, we are able to compare the effect of different IMFs on multiple observables and derived quantities in astrophysics. Specifically, we consider a non-universal IMF that varies as a function of the local star formation rate, and explore the effects on the star formation rate density (SFRD), the extragalactic background light, the supernova (both core-collapse and thermonuclear) rates, and the diffuse supernova neutrino background. Our most interesting result is that our adopted varying IMF leads to much greater uncertainty on the SFRD at $z \approx 2-4$ than is usually assumed. Indeed, we find an SFRD (inferred using observed galaxy luminosity distributions) that is a factor of $\gtrsim 3$ lower than canonical results obtained using a universal IMF. Secondly, the non-universal IMF we explore implies a reduction in the supernova core-collapse rate of a factor of $\sim 2$, compared against a universal IMF. The other potential tracers are only slightly affected by changes to the properties of the IMF. We find that currently available data do not provide a clear preference for universal or non-universal IMF. However, improvements to measurements of the star formation rate and core-collapse supernova rate at redshifts $z \gtrsim 2$ may offer the best prospects for discernment.