When the upper layer of an accreting neutron star experiences a thermonuclear runaway of helium and hydrogen, it exhibits an X-ray burst of a few keV with a cool-down phase of typically 1 minute. When there is a surplus of hydrogen, hydrogen fusion is expected to simmer during that same minute due to the rp process, which consists of rapid proton captures and slow {beta}-decays of proton-rich isotopes. We have analyzed the high-quality light curves of 1254 X-ray bursts, obtained with the Proportional Counter Array on the Rossi X-ray Timing Explorer between 1996 and 2012, to systematically study the cooling and rp process. This is a follow-up of a study on a selection of 37 bursts from systems that lack hydrogen and show only cooling during the bursts.We find that the bolometric light curves are well described by the combination of a power law and a one-sided Gaussian. The power-law decay index is between 1.3 and 2.1 and similar to that for the 37-bursts sample. There are individual bursters with a narrower range. The Gaussian is detected in half of all bursts, with a typical standard deviation of 50s and a fluence ranging up to 60% of the total fluence. The Gaussian appears consistent with being due to the rp process. The Gaussian fluence fraction suggests that the layer where the rp process is active is underabundant in H by a factor of at least five with respect to cosmic abundances. Ninety-four percent of all bursts from ultracompact X-ray binaries lack the Gaussian component, and the remaining 6% are marginal detections. This is consistent with a hydrogen deficiency in these binaries. We find no clear correlation between the power law and Gaussian light-curve components.