  A study of the thermal properties of two-dimensional topological lattice
models is presented. This work is relevant to assess the usefulness of these
systems as a quantum memory. For our purposes, we use the topological mutual
information $I_{\mathrm{topo}}$ as a "topological order parameter". For Abelian
models, we show how $I_{\mathrm{topo}}$ depends on the thermal topological
charge probability distribution. More generally, we present a conjecture that
$I_{\mathrm{topo}}$ can (asymptotically) be written as a Kullback-Leitner
distance between this probability distribution and that induced by the quantum
dimensions of the model at hand. We also explain why $I_{\mathrm{topo}}$ is
more suitable for our purposes than the more familiar entanglement entropy
$S_{\mathrm{topo}}$. A scaling law, encoding the interplay of volume and
temperature effects, as well as different limit procedures, are derived in
detail. A non-Abelian model is next analysed and similar results are found.
Finally, we also consider, in the case of a one-plaquette toric code, an
environment model giving rise to a simulation of thermal effects in time.
