  The variable activity of a gamma-ray burst (GRB) source is thought to be
correlated with its absolute peak luminosity - a relation that, if confirmed,
can be used to derive an independent estimate of the redshift of a GRB. We find
that bursts with highly variable light curves have greater nu-F_nu spectral
peak energies, when we transform these energies to the cosmological rest frame
using the redshift estimates derived either from optical spectral features or
from the luminosity-variability distance indicator itself. This positive
correlation between peak energy and variability spans ~ 2 orders of magnitude
and appears to accommodate GRB 980425, lending credibility to the association
of this burst with SN 1998bw. The existence of such a correlation not only
provides an interesting clue to the nature of this luminosity indicator but
potentially reinforces the validity of the redshift estimates derived from this
method. It also implies that the rest frame GRB peak energy is correlated with
the intrinsic luminosity of the burst, as has been suggested in the past as an
explanation of the observed hardness-intensity correlation in GRBs.
