  We generalize the Ensemble Geometric Phase (EGP), recently introduced to
classify the topology of density matrices, to finite-temperature states of
interacting systems in one spatial dimension (1D). This includes cases where
the gapped ground state has a fractional filling and is degenerate. At zero
temperature the corresponding topological invariant agrees with the well-known
invariant of Niu, Thouless and Wu. We show that its value at finite
temperatures is identical to that of the ground state below some critical
temperature $T_c$ larger than the many-body gap. We illustrate our result with
numerical simulations of the 1D extended super-lattice Bose-Hubbard model at
quarter filling. Here a cyclic change of parameters in the ground state leads
to a topological charge pump with fractional winding $\nu=1/2$. The particle
transport is no longer quantized when the temperature becomes comparable to the
many-body gap, yet the winding of the generalized EGP is.
