  We discuss in detail the broad band X-ray variability expected in Seyfert
galaxies in the framework of a Compton cooled corona model. Variations in the
optical depth tau of the corona cause spectral changes in the Comptonized
emission in the sense that the spectrum steepens and the temperature decreases
with increasing tau. If the corona is pair dominated, tau is determined by the
compactness (proportional to luminosity for fixed size). This yields a definite
relation between spectral shape and intensity implying a variation in intensity
of a factor 10 for a change of 0.2 in the 2-10 keV spectral index. For low pair
density coronae, no definite prediction is possible. In the absence of
substantial changes in the structure of the corona, the soft thermal component
correlates with the medium-hard Comptonized component. Examples are computed
using the ROSAT response matrix, showing that current observations of NGC 5548
and Mrk 766 showing different behaviours can be reconciled with the model. The
spectral behaviour of Mrk 766 strongly suggests that the coronal plasma in this
source is not pair dominated.
