  Two-dimensional topological phases are characterized by TKNN integers, which
classify Bloch energy bands or groups of Bloch bands. However, quantization
does not survive thermal averaging or dephasing to mixed states. We show that
using Uhlmann's parallel transport for density matrices (Rep. Math. Phys. 24,
229 (1986)), an integer classification of topological phases can be defined for
a finite generalized temperature $T$ or dephasing Lindbladian. This scheme
reduces to the familiar TKNN classification for $T<T_{{\rm c},1}$, becomes
trivial for $T>T_{{\rm c},2}$, and exhibits a `gapless' intermediate regime
where topological indices are not well-defined. We demonstrate these ideas in
detail, applying them to Haldane's honeycomb lattice model and the
Bernevig-Hughes-Zhang model, and we comment on their generalization to
multi-band Chern insulators.
