  Topological order has become a new paradigm to distinguish ground states of
interacting many-body systems without conventional long-range order. Here we
discuss possible extensions of this concept to density matrices describing
statistical ensembles. For a large class of quasi-thermal states, which can be
realized as thermal states of some quasi-local Hamiltonian, we generalize
earlier definitions of density matrix topology to generic many-body systems
with strong correlations. We point out that the robustness of topological
order, defined as a pattern of long-range entanglement, depends crucially on
the perturbations under consideration. While it is intrinsically protected
against local perturbations of arbitrary strength in an infinite closed quantum
system, purely local perturbations can destroy topological order in open
systems coupled to baths if the coupling is sufficiently strong. We discuss our
classification scheme using the finite-temperature quantum Hall states and
point out that the classical Hall effect can be understood as a finite
temperature topological phase.
