We study the spectral evolution on second and sub--second timescales in 11 long and 12 short Gamma Ray Bursts (GRBs) with peak flux >8.5e-6 erg/cm2 s (8 keV-35 MeV) detected by the Fermi satellite. The peak flux correlates with the time-averaged peak energy in both classes of bursts. The peak energy evolution, as a function of time, tracks the evolution of the flux on short timescales in both short and long GRBs. We do not find evidence of an hard-to-soft spectral evolution. While short GRBs have observed peak energies larger than few MeV during most of their evolution, long GRBs can start with a softer peak energy (of few hundreds keV) and become as hard as short ones (i.e. with Ep,obs larger than few MeV) at the peak of their light curve. Six GRBs in our sample have a measured redshift. In these few cases we find that their correlations between the rest frame Ep and the luminosity Liso are less scattered than their correlations in the observer frame between the peak energy Ep,obs and the flux P. We find that the rest frame Ep of long bursts can be as high or even larger than that of short GRBs and that short and long GRBs follow the same Ep-Liso correlation, despite the fact that they likely have different progenitors.
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