  The reservoir of molecular gas (H$_{\rm 2}$) represents the fuel for the star
formation (SF) of a galaxy. Connecting the star formation rate (SFR) to the
available H$_{\rm 2}$ is key to accurately model SF in cosmological simulations
of galaxy formation. We investigate how modifying the underlying modelling of
H$_{\rm 2}$ and the description of stellar feedback in low-metallicity
environments (LMF, i.e. low-metallicity stellar feedback) in cosmological,
zoomed-in simulations of a Milky Way-size halo influences the formation history
of the forming, spiral galaxy and its final properties. We exploit two
different models to compute the molecular fraction of cold gas (f$_{\rm H_{\rm
2}}$): $i)$ the theoretical model by Krumholz et al. (2009b) and $ii)$ the
phenomenological prescription by Blitz & Rosolowsky (2006). We find that the
model adopted to estimate f$_{\rm H_{\rm 2}}$ plays a key role in determining
final properties and in shaping the morphology of the galaxy. The clumpier
interstellar medium (ISM) and the more complex H$_{\rm 2}$ distribution that
the Krumholz et al. (2009b) model predicts result in better agreement with
observations of nearby disc galaxies. This shows how crucial it is to link the
SFR to the physical properties of the star-forming, molecular ISM. The
additional source of energy that LMF supplies in a metal-poor ISM is key in
controlling SF at high redshift and in regulating the reservoir of SF across
cosmic time. Not only is LMF able to regulate cooling properties of the ISM,
but it also reduces the stellar mass of the galaxy bulge. These findings can
foster the improvement of the numerical modelling of SF in cosmological
simulations.
