The Universe's largest galaxies reside at the centres of galaxy clusters and are embedded in hot gas that, if left undisturbed, would cool quickly and create many more new stars than are actually observed(1-5). Cooling can be regulated by feedback from accretion of cooling gas onto the central black hole, but requires an accretion rate finely tuned to the thermodynamic state of the hot gas(6,7). Theoretical models in which cold clouds precipitate out of the hot gas via thermal instability and accrete onto the black hole exhibit the necessary tuning(8-10). Recent observational evidence shows that the abundance of cold gas in the centres of clusters increases rapidly near the predicted threshold for instability(11). Here we report observations showing that this precipitation threshold extends over a large range in cluster radius, cluster mass and cosmic time. We incorporate the precipitation threshold into a framework of theoretical models for the thermodynamic state of hot gas in galaxy clusters. According to that framework, precipitation regulates star formation in some giant galaxies, while thermal conduction prevents star formation in others if it can compensate for radiative cooling and shut off precipitation.