  We consider the quantum XY model and study the effects of interacting
perturbations on the time evolution of the von Neumann and R\'enyi entropies of
spin blocks after global quenches. We show that the entropies are sensitive to
perturbations that break hidden symmetries behind the integrability of the
model. At times much larger than the characteristic time of the well-known
linear increase of the entropies, we identify a time window characterized by a
novel linear growth followed by saturation. The typical time of the phenomenon
is inversely proportional to the perturbation strength and the behavior is
trigger off by the extinction of an infinite number of local conservation laws
following a non-abelian algebra. The universality of the crossover is revealed
by a semi-classical picture that captures the leading behavior of the
entropies. We check our theoretical predictions against iTEBD simulations. The
good agreement between theory and numerics substantiates the method developed
in [Bertini and Fagotti, J. Stat. Mech. (2015) P07012] for investigating a
pre-relaxation limit in weakly interacting models.
