  Employing two samples containing of 56 and 59 well-separated FRED (fast rise
and exponential decay) gamma-ray burst (GRB) pulses whose spectra are fitted by
the Band spectrum and Compton model, respectively, we have investigated the
evolutionary slope of $E_{p}$ (where $E_{p}$ is the peak energy in the $\nu
F\nu$ spectrum) with time during the pulse decay phase. The bursts in the
samples were observed by the Burst and Transient Source Experiment (BATSE) on
the Compton Gamma-Ray Observatory. We first test the $E_{p}$ evolutionary slope
during the pulse decay phase predicted by Lu et al. (2007) based on the model
of highly symmetric expanding fireballs in which the curvature effect of the
expanding fireball surface is the key factor concerned. It is found that the
evolutionary slopes are normally distributed for both samples and concentrated
around the values of 0.73 and 0.76 for Band and Compton model, respectively,
which is in good agreement with the theoretical expectation of Lu et al.
(2007). However, the inconsistence with their results is that the intrinsic
spectra of most of bursts may bear the Comptonized or thermal synchrotron
spectrum, rather than the Band spectrum. The relationships between the
evolutionary slope and the spectral parameters are also checked. We show the
slope is correlated with $E_{p}$ of time-integrated spectra as well as the
photon flux but anticorrelated with the lower energy index $\alpha$. In
addition, a correlation between the slope and the intrinsic $E_{p}$ derived by
using the pseudo-redshift is also identified. The mechanisms of these
correlations are unclear currently and the theoretical interpretations are
required.
