  The paper is a continuation of work [15] in which the general setting for
analogs of the Szeg\"o theorem for ergodic operators was given and several
interesting cases were considered. Here we extend the results of [15] to a
wider class of test functions and symbols which determine the Szeg\"o-type
asymptotic formula for the one-dimensional Schrodinger operator with random
potential. We show that in this case the subleading term of the formula is
given by a Central Limit Theorem in the spectral context, hence the term is
asymptotically proportional to $L^{1/2}$, where $L$ is the length of the
interval on which the Schrodinger operator is initially defined. This has to be
compared with the classical Szeg\"o formula, where the subleading term is
bounded in $L$, $L \to \infty$. We prove an analog of standard Central Limit
Theorem (the convergence of the probability of the corresponding event to the
Gaussian Law) as well as an analog of the almost sure Central Limit Theorem
(the convergence with probability 1 of the logarithmic means of the indicator
of the corresponding event to the Gaussian Law). We illustrate our general
results by establishing the asymptotic formula for the entanglement entropy of
free disordered Fermions for non-zero temperature.
