We have investigated the evolution of observed spectral hardness E p (where E p is the maximum of the F spectrum) based on the model of highly symmetric expanding fireballs, where the Doppler effect of the expanding fireball surface is the key factor concerned, and find that the evolutionary curve of E p undergoes a drop to rise to decay evolution (the A, B, and C phases, respectively). The A phase peaks at the very onset of the corresponding light-curve pulse. Since the Doppler effect is less prominent in the B phase, corresponding to the rising portion of the pulse, the slope of the B phase reflects both the corresponding intrinsic spectral evolution and the shape of its local pulse. The C phase, corresponding to the decay portion of the pulse, where the Doppler effect dominates, always decreases monotonically and does not necessarily reflect the corresponding intrinsic spectral evolution. Investigation of a sample of 12 FRED pulses (their peak energy can be found in the work of Kaneko et al.) shows that the evolutionary characteristics of the fitted peak energy in the FRED pulses are in good agreement with our predictions.