  We study quantum quenches in the two-dimensional Kitaev toric code model and
compute exactly the time-dependent entanglement entropy of the non-equilibrium
wave-function evolving from a paramagnetic initial state with the toric code
Hamiltonian. We find that the area law survives at all times. Adding disorder
to the toric code couplings makes the entanglement entropy per unit boundary
length saturate to disorder-independent values at long times and in the
thermodynamic limit. There are order-one corrections to the area law from the
corners in the subsystem boundary but the topological entropy remains zero at
all times. We argue that breaking the integrability with a small magnetic field
could change the area law to a volume scaling as expected of thermalized states
but is not sufficient for forming topological entanglement due to the presence
of an excess energy and a finite density of defects.
