  Topology in many-body physics usually emerges as a feature of equilibrium
quantum states. We show that topological fingerprints can also appear in the
relaxation rates of open quantum systems. To demonstrate this we consider one
of the simplest models that has two distinct topological phases in its ground
state: the Kitaev model for the $p$-wave superconductor. After introducing
dissipation to this model we estimate the Liouvillian gap in both strong and
weak dissipative limits. Our results show that a non-zero superconducting
pairing opens a Liouvillian gap that remains open in the limit of infinite
system size. At strong dissipation this gap is essentially unaffected by the
topology of the underlying Hamiltonian ground state. In contrast, when
dissipation is weak, the topological phase of the Hamiltonian ground state
plays a crucial role in determining the character of the Liouvillian gap. We
find, for example, that in the topological phase this gap is completely immune
to changes in the chemical potential. On the other hand, in the non-topological
phase the Liouvillian gap is suppressed by a large chemical potential.
