  The leading asymptotic large-scale behaviour of the spatially bipartite
entanglement entropy (EE) of the free Fermi gas infinitely extended in
multidimensional Euclidean space at zero absolute temperature, T=0, is by now
well understood. Here, we present and discuss the first rigorous results for
the corresponding EE of thermal equilibrium states at T>0. The leading
large-scale term of this thermal EE turns out to be twice the first-order
finite-size correction to the infinite-volume thermal entropy (density). Not
surprisingly, this correction is just the thermal entropy on the interface of
the bipartition. However, it is given by a rather complicated integral derived
from a semiclassical trace formula for a certain operator on the underlying
one-particle Hilbert space. But in the zero-temperature limit the leading
large-scale term of the thermal EE considerably simplifies and displays a
ln(1/T)-singularity which one may identify with the known logarithmic
enhancement at T=0 of the so-called area-law scaling.
