  Understanding the behaviour of topologically ordered lattice systems at
finite temperature is a way of assessing their potential as fault-tolerant
quantum memories. We compute the natural extension of the topological
entanglement entropy for T > 0, namely the subleading correction
$I_{\textrm{topo}}$ to the area law for mutual information. Its dependence on T
can be written, for Abelian Kitaev models, in terms of information-theoretic
functions and readily identifiable scaling behaviour, from which the interplay
between volume, temperature, and topological order, can be read. These
arguments are extended to non-Abelian quantum double models, and numerical
results are given for the $D(S_3)$ model, showing qualitative agreement with
the Abelian case.
