Observations of star-forming galaxies at high z have suggested discrepancies in the inferred star formation rates (SFRs) either between data and models or between complementary measures of the SFR. These putative discrepancies could all be alleviated if the stellar initial mass function (IMF) is systematically weighted towards more high-mass star formation in rapidly star-forming galaxies. Here, we explore how the IMF might vary under the central assumption that the turnover mass in the IMF, , scales with the Jeans mass in giant molecular clouds (GMCs), . We employ hydrodynamic simulations of galaxies coupled with radiative transfer models to predict how the typical GMC Jeans mass, and hence the IMF, varies with galaxy properties. We then study the impact of such an IMF on the star formation law, the SFR–M* relation, sub-millimetre galaxies (SMGs) and the cosmic SFR density. Our main results are: the H2 mass-weighted Jeans mass in a galaxy scales well with the SFR when the SFR is greater than a few M⊙ yr−1. Stellar population synthesis modelling shows that this results in a non-linear relation between SFR and Lbol, such that SFR . Using this model relation, the inferred SFR of local ultraluminous infrared galaxies decreases by a factor of ∼2, and that of high-z SMGs decreases by a factor of ∼3–5. At z∼ 2, this results in a lowered normalization of the SFR–M* relation in better agreement with models, a reduced discrepancy between the observed cosmic SFR density and stellar mass density evolution, and SMG SFRs that are easier to accommodate in current hierarchical structure formation models. It further results in a Kennicutt–Schmidt star formation law with a slope of ∼1.6 when utilizing a physically motivated form for the CO–H2 conversion factor that varies with galaxy physical property. While each of the discrepancies considered here could be alleviated without appealing to a varying IMF, the modest variation implied by assuming is a plausible solution that simultaneously addresses numerous thorny issues regarding the SFRs of high-z galaxies.