In quantum physics, topological properties usually emerge as a feature of equilibrium quantum states. We show that topological fingerprints can also manifest 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 regimes. Our results show that a non-zero superconducting pairing opens a Liouvillian gap that remains open at large system sizes. 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 present the method used to extract this Liouvillian gap for a number of different dissipative processes.