Recent studies proposed that cosmic rays (CR) are a key ingredient in setting the conditions for star formation, thanks to their ability to alter the thermal and chemical state of dense gas in the UV-shielded cores of molecular clouds. In this paper, we explore their role as regulators of the stellar initial mass function (IMF) variations, using the semi-analytic model for GAlaxy Evolution and Assembly (GAEA). The new model confirms our previous results obtained using the integrated galaxy-wide IMF (IGIMF) theory: both variable IMF models reproduce the observed increase of $alpha$-enhancement as a function of stellar mass and the measured $z=0$ excess of dynamical mass-to-light ratios with respect to photometric estimates assuming a universal IMF. We focus here on the mismatch between the photometrically-derived ($M^{rm app}_{star}$) and intrinsic ($M_{star}$) stellar masses, by analysing in detail the evolution of model galaxies with different values of $M_{star}/M^{rm app}_{star}$. We find that galaxies with small deviations (i.e. formally consistent with a universal IMF hypothesis) are characterized by more extended star formation histories and live in less massive haloes with respect to the bulk of the galaxy population. While the IGIMF theory does not change significantly the mean evolution of model galaxies with respect to the reference model, a CR-regulated IMF implies shorter star formation histories and higher peaks of star formation for objects more massive than $10^{10.5} M_odot$. However, we also show that it is difficult to unveil this behaviour from observations, as the key physical quantities are typically derived assuming a universal IMF.
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