  We analyze the entanglement properties of the asymptotic steady state after a
quench from free to hard-core bosons in one dimension. The R\'enyi and von
Neumann entanglement entropies are found to be extensive, and the latter
coincides with the thermodynamic entropy of the Generalized Gibbs Ensemble
(GGE). Computing the spectrum of the two-point function, we provide exact
analytical results both for the leading extensive parts and the subleading
terms for the entropies as well as for the cumulants of the particle number
fluctuations. We also compare the extensive part of the entanglement entropy
with the thermodynamic ones, showing that the GGE entropy equal the
entanglement one and it is the double of the diagonal entropy.
