ABSTRACT We determine the cosmic abundance of molecular hydrogen (H2) in the local Universe from the xCOLD GASS survey. To constrain the H2 mass function at low masses and correct for the effect of the lower stellar mass limit of $10^9 \, \mathrm{M}_{\odot }$ in the xCOLD GASS survey, we use an empirical approach based on an observed scaling relation between star formation rate and gas mass. We also constrain the H i and H i+H2 mass functions using the xGASS survey and compare them to the H i mass function from the ALFALFA survey. We find the cosmic abundance of molecular gas in the local Universe to be $\Omega _{\mathrm{H_2}} = (5.34 \pm 0.47) \times 10^{-5} h^{-1}$. Molecular gas accounts for $19.6\pm 3.9{{\ \rm per\ cent}}$ of the total abundance of cold gas, $\mathrm{\Omega _{H\,{\small I}+H_{2}}} = (4.66 \pm 0.70) \times 10^{-4}\, h_{70}^{-1}$. Galaxies with stellar masses in excess of 109$\, \mathrm{M}_{\odot }$ account for 89 per cent of the molecular gas in the local Universe, while in comparison such galaxies only contain 73 per cent of the cold atomic gas as traced by the H i 21cm line. The xCOLD GASS CO, molecular gas and cold gas mass functions, and $\Omega _{\mathrm{H_2}}$ measurements provide constraints for models of galaxy evolution and help to anchor blind molecular gas surveys attempting to determine the abundance of molecular gas at high redshifts.