Topology in open many-body systems

Topological states of matter have fascinated physicists since a long time. The notion of topology is however ususally associated with ground states of (many-body)-Hamiltonians, which are pure. So what is left of it at finite temperatures and can topological protection be extended to non-equilibrium steady states (NESS) of open systems? Can suitable observables be constructed that preserve the integrity of topological invariants for mixed states and what are measurable consequences of their existence? Can we classify the topology of finite-temperature and NESS using generalized symmetries?