THE nature of gamma-ray bursts remains one of the great mysteries of modern astromomy(1,2). Although the spatial distribution of these high-energy sources is tightly constrained (they are distributed isotropically across the sky(3)), the same cannot be said of their spectral and temporal properties, for which few clear trends have been identified(1,2). But it is from such properties that important clues about the physical mechanisms underlying the bursts are likely to be derived(4,5). Here we show that for a sample of bursts consisting of well resolved, isolated energy pulses, and for which the spectra are sufficiently clean to allow their temporal evolution to be modelled, the photon energy at which the power output is maximum for each pulse decreases exponentially with photon fluence (that is, running time integral of photon flux), We find that for many multi-pulse bursts, the exponential decay constant is invariant from pulse to pulse. This property favours models in which the pulses arise from a regenerative source, rather than as a consequence of a single catastrophic event.