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The recently inferred variations in the stellar initial mass function (IMF)among local high-mass early-type galaxies may require a reinterpretation ofobservations of galaxy populations and may have important consequences for thepredictions of models of galaxy formation and evolution. We present a new pairof cosmological, hydrodynamical simulations based on the EAGLE model thatself-consistently adopt an IMF that respectively becomes bottom- or top-heavyin high-pressure environments for individual star-forming gas particles. Insuch models, the excess stellar mass-to-light ($M/L$) ratio with respect to areference IMF is increased due to an overabundance of low-mass dwarf stars orstellar remnants, respectively. Crucially, both pressure-dependent IMFs havebeen calibrated to reproduce the observed trends of increasing excess $M/L$with central stellar velocity dispersion ($\sigma_e$) in early-type galaxies,while maintaining agreement with the observables used to calibrate the EAGLEmodel, namely the galaxy luminosity function, half-light radii of late-typegalaxies, and black hole masses. We find that while the $M/L$ excess is a goodmeasure of the IMF for low-mass slope variations, it depends strongly on theage of the stellar population for high-mass slope variations. The normalizationof the [Mg/Fe]$-\sigma_e$ relation is decreased (increased) for bottom-(top-)heavy IMF variations, while the slope is not strongly affected.Bottom-heavy variations have little impact on galaxy metallicities, half-lightradii of early-type galaxies, or star formation rates, while top-heavyvariations significantly increase these quantities for high-mass galaxies,leading to tension with observations.

Calibrated, cosmological hydrodynamical simulations with variable IMFs I: Method and effect on global galaxy scaling relations