Supermassive black holes, with masses more than a million times that of the Sun, seem to inhabit the centres of all massive galaxies. Cosmologically motivated theories of galaxy formation require feedback from these supermassive black holes to regulate star formation. In the absence of such feedback, state-of-the-art numerical simulations fail to reproduce the number density and properties of massive galaxies in the local Universe. There is, however, no observational evidence of this strongly coupled coevolution between supermassive black holes and star formation, impeding our understanding of baryonic processes within galaxies. Here we report that the star formation histories of nearby massive galaxies, as measured from their integrated optical spectra, depend on the mass of the central supermassive black hole. Our results indicate that the black-hole mass scales with the gas cooling rate in the early Universe. The subsequent quenching of star formation takes place earlier and more efficiently in galaxies that host higher-mass central black holes. The observed relation between black-hole mass and star formation efficiency applies to all generations of stars formed throughout the life of a galaxy, revealing a continuous interplay between black-hole activity and baryon cooling.