At the highest stellar masses (log(M*)>~11.5M_{sun}_), only a small fraction of galaxies are disk-like and actively star-forming objects. These so-called `super spirals' are ideal objects to better understand how galaxy evolution proceeds and to extend our knowledge about the relation between stars and gas to a higher stellar mass regime. We present new CO(1-0) data for a sample of 46 super spirals and for 18 slightly lower-mass (log((M*)>11.0M_{sun}_) galaxies with broad HI lines -- HI fast-rotators (HI-FRs). We analyze their molecular gas mass, derived from CO(1-0), in relation to their star formation rate (SFR) and stellar mass, and compare the results to values and scaling relations derived from lower-mass galaxies. We confirm that super spirals follow the same star-forming main sequence (SFMS) as lower-mass galaxies. We find that they possess abundant molecular gas (mean redshift-corrected molecular gas mass fraction (log(f_mol,zcorr_)=-1.36+/-0.02), which lies above the extrapolation of the scaling relation with stellar mass derived from lower-mass galaxies, but within the relation between f_mol_ and the distance to the SFMS. The molecular gas depletion time, {tau}_dep_=M_mol_/SFR, is higher than for lower-mass galaxies on the SFMS ({tau}_dep_=9.30+/-0.03, compared to {tau}_dep_=9.00+/-0.02 for the comparison sample) and seems to continue an increasing trend with stellar mass. HI-FR galaxies have an atomic-to-molecular gas mass ratio that is in agreement with that of lower-mass galaxies, indicating that the conversion from the atomic to molecular gas proceeds in a similar way. We conclude that the availability of molecular gas is a crucial factor to enable star formation to continue and that, if gas is present, quenching is not a necessary destiny for high-mass galaxies. The difference in gas depletion time suggests that the properties of the molecular gas at high stellar masses are less favorable for star formation.