  Aims: We used data from the 58 month long, continuous Swift/Burst Alert
Telescope (BAT) observations of the five brightest Seyfert galaxies at hard
X-rays, to study their flux and spectral variability in the 20-100 keV energy
band. The column density in these objects is less than 10^24 cm-2, which
implies that the Swift/BAT data allow us to study the "true" variability of the
central source. Results: All objects show significant variations, with an
amplitude which is similar to the AGN variability amplitude at energies below
10 keV. We found evidence for an anti-correlation between variability amplitude
and black hole mass. The light curves in both bands are well correlated, with
no significant delays on time scales as short as 2 days. NGC 4151 and NGC 2110
do not show spectral variability, but we found a significant anti-correlation
between hardness ratios and source flux in NGC 4388 (and NGC 4945, IC 4329, to
a lesser extent). This "softer when brighter" behaviour is similar to what has
been observed at energies below 10 keV, and cannot be explained if the
continuum varies only in flux; the intrinsic shape should also steepen with
increasing flux. Conclusions: The presence of significant flux variations
indicate that the central source in these objects is intrinsically variable on
time scales as short as 1-2 days. The intrinsic slope of the continuum varies
with the flux (at least in NGC 4388). The positive "spectral slope-flux"
correlation can be explained if the temperature of the hot corona decreases
with increasing flux. The lack of spectral variations in two objects, could be
due to the fact that they may operate in a different "state", as their
accretion rate is less than 1% of the Eddington limit (significantly smaller
than the rate of the other three objects in the sample).
