  We construct a topological invariant that classifies density matrices of
symmetry-protected topological orders in two-dimensional fermionic systems. As
it is constructed out of the previously introduced Uhlmann phase, we refer to
it as the topological Uhlmann number ${\rm n}_{\rm U}$. With it, we study
thermal topological phases in several two-dimensional models of topological
insulators and superconductors, computing phase diagrams where the temperature
$T$ is on an equal footing with the coupling constants in the Hamiltonian.
Moreover, we find novel thermal-topological transitions between two non-trivial
phases in a model with high Chern numbers. At small temperature we recover the
standard topological phases as the Uhlmann number approaches to the Chern
number.
