  We consider the discrete defocusing nonlinear Schr\"odinger equation in its
integrable version, which is called defocusing Ablowitz-Ladik lattice. We
consider periodic boundary conditions with period $N$ and initial data sample
according to the Generalized Gibbs ensemble. In this setting, the Lax matrix of
the Ablowitz-Ladik lattice is a random CMV-periodic matrix and it is related to
the Killip-Nenciu Circular $\beta$-ensemble at high-temperature. We obtain the
generalized free energy of the Ablowitz-Ladik lattice and the density of states
of the random Lax matrix by establishing a mapping to the one-dimensional
log-gas. For the Gibbs measure related to the Hamiltonian of the Ablowitz-Ladik
flow, we obtain the density of states via a particular solution of the
double-confluent Heun equation.
