Recent observational and theoretical studies indicate that the stellar initial mass function (IMF) varies systematically with the environment (star formation rate - SFR, metallicity). Although the exact dependence of the IMF on those properties is likely to change with improving observational constraints, the reported trend in the shape of the IMF appears robust. We present the first study aiming to evaluate the effect of the IMF variations on the measured cosmic SFR density (SFRD) as a function of metallicity and redshift, fSFR(Z, z). We also study the expected number and metallicity of white dwarf, neutron star, and black hole progenitors under different IMF assumptions. Applying the empirically driven IMF variations described by the integrated galactic IMF (IGIMF) theory, we revise fSFR(Z, z) obtained in our previous study that assumed a universal IMF. We find a lower SFRD at high redshifts and a higher fraction of metal-poor stars being formed than previously determined. In the local Universe, our calculation applying the IGIMF theory suggests more white dwarf and neutron star progenitors in comparison with the universal IMF scenario, while the number of black hole progenitors remains unaffected.