  Concepts of information theory are increasingly used to characterize
collective phenomena in condensed matter systems, such as the use of
entanglement entropies to identify emergent topological order in interacting
quantum many-body systems. Here we employ classical variants of these concepts,
in particular R\'enyi entropies and their associated mutual information, to
identify topological order in classical systems. Like for their quantum
counterparts, the presence of topological order can be identified in such
classical systems via a universal, subleading contribution to the prevalent
volume and boundary laws of the classical R\'enyi entropies. We demonstrate
that an additional subleading $O(1)$ contribution generically arises for all
R\'enyi entropies $S^{(n)}$ with $n \geq 2$ when driving the system towards a
phase transition, e.g. into a conventionally ordered phase. This additional
subleading term, which we dub connectivity contribution, tracks back to partial
subsystem ordering and is proportional to the number of connected parts in a
given bipartition. Notably, the Levin-Wen summation scheme -- typically used to
extract the topological contribution to the R\'enyi entropies -- does not fully
eliminate this additional connectivity contribution in this classical context.
This indicates that the distillation of topological order from R\'enyi
entropies requires an additional level of scrutiny to distinguish topological
from non-topological $O(1)$ contributions. This is also the case for quantum
systems, for which we discuss which entropies are sensitive to these
connectivity contributions. We showcase these findings by extensive numerical
simulations of a classical variant of the toric code model, for which we study
the stability of topological order in the presence of a magnetic field and at
finite temperatures from a R\'enyi entropy perspective.
