  The ground states of noninteracting fermions in one-dimension with chiral
symmetry form a class of topological band insulators, described by a
topological invariant that can be related to the Zak phase. Recently, a
generalization of this quantity to mixed states - known as the ensemble
geometric phase (EGP) - has emerged as a robust way to describe topology at
non-zero temperature. By using this quantity, we explore the nature of topology
allowed for dissipation beyond a Lindblad description, to allow for coupling to
external baths at finite temperatures. We introduce two main aspects to the
theory of mixed state topology. First, we discover topological phase
transitions as a function of the temperature T, manifesting as changes in
winding number of the EGP accumulated over a closed loop in parameter space. We
characterize the nature of these transitions and reveal that the corresponding
non-equilibrium steady state at the transition can exhibit a nontrivial
structure - contrary to previous studies where it was found to be in a fully
mixed state. Second, we demonstrate that the EGP itself becomes quantized when
key symmetries are present, allowing it to be viewed as a topological marker
which can undergo equilibrium topological transitions at non-zero temperatures.
