Abstract We update our recently published model for GAlaxy Evolution and Assembly (GAEA), to include a self-consistent treatment of the partition of cold gas in atomic and molecular hydrogen. Our model provides significant improvements with respect to previous ones used for similar studies. In particular, GAEA (i) includes a sophisticated chemical enrichment scheme accounting for non-instantaneous recycling of gas, metals and energy; (ii) reproduces the measured evolution of the galaxy stellar mass function; (iii) reasonably reproduces the observed correlation between galaxy stellar mass and gas metallicity at different redshifts. These are important prerequisites for models considering a metallicity-dependent efficiency of molecular gas formation. We also update our model for disc sizes and show that model predictions are in nice agreement with observational estimates for the gas, stellar and star-forming discs at different cosmic epochs. We analyse the influence of different star formation laws including empirical relations based on the hydrostatic pressure of the disc, analytic models and prescriptions derived from detailed hydrodynamical simulations. We find that modifying the star formation law does not affect significantly the global properties of model galaxies, neither their distributions. The only quantity showing significant deviations in different models is the cosmic molecular-to-atomic hydrogen ratio, particularly at high redshift. Unfortunately, however, this quantity also depends strongly on the modelling adopted for additional physical processes. Useful constraints on the physical processes regulating star formation can be obtained focusing on low-mass galaxies and/or at higher redshift. In this case, self-regulation has not yet washed out differences imprinted at early time.