We have analyzed the evolution of the spectral hardness parameter Epk (the maximum of the νFν spectrum) as a function of fluence in gamma-ray bursts. We fit 41 pulses within 26 bursts with the trend reported by Liang & Kargatis, which found that Epk decays exponentially with respect to photon fluence Φ(t). We also fit these pulses with a slight modification of this trend, where Epk decays linearly with energy fluence. In both cases, we found the set of 41 pulses to be consistent with the trend. For the latter trend, which we believe to be more physical, the distribution of the decay constant Φ0 is roughly lognormal, where the mean of log10 Φ0 is 1.75 ± 0.07 and the FWHM of log10 Φ0 is 1.0 ± 0.1. Regarding an earlier reported invariance in Φ0 among different pulses in a single burst, we found probabilities of 0.49-0.84 (depending on the test used) that such invariance would occur by coincidence, most likely because of the narrow distribution of Φ0 values among pulses.
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