Abstract We present single-dish 12CO(1–0) and 12CO(2–1) observations for 14 low-redshift quasi-stellar objects (QSOs). In combination with optical integral field spectroscopy, we study how the cold gas content relates to the star formation rate (SFR) and black hole accretion rate. 12CO(1–0) is detected in 8 of 14 targets and 12CO(2–1) is detected in 7 out of 11 cases. The majority of disc-dominated QSOs reveal gas fractions and depletion times matching normal star-forming systems. Two gas-rich major mergers show clear starburst signatures with higher than average gas fractions and shorter depletion times. Bulge-dominated QSO hosts are mainly undetected in 12CO(1–0), which corresponds, on average, to lower gas fractions than in disc-dominated counterparts. Their SFRs, however, imply shorter than average depletion times and higher star formation efficiencies. Negative QSO feedback through removal of cold gas seems to play a negligible role in our sample. We find a trend between black hole accretion rate and total molecular gas content for disc-dominated QSOs when combined with literature samples. We interpret this as an upper envelope for the nuclear activity and it is well represented by a scaling relation between the total and circumnuclear gas reservoir accessible for accretion. Bulge-dominated QSOs significantly differ from that scaling relation and appear uncorrelated with the total molecular gas content. This could be explained either by a more compact gas reservoir, blown out of the gas envelope through outflows, or a different interstellar medium phase composition.