  We study the thermodynamical properties of the quantum kicked rotator,
coarsened by an external fluctuation with a weak intensity D, by means of the
Tsallis entropy with a changing entropic index q. The genuine entropic index,
corresponding to given values of D and $\hbar$ is that making the Tsallis
entropy increase linearly in time, and it is proved to become q <1 for suitably
large values of $\hbar $: This indicates a subdiffusional regime which, in
turn, signals the occurrence of quantum localization. Thus the process of
Anderson localization is shown to be compatible with a thermodynamical
representation provided that a non-extensive form of entropy is used.
