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.Comment: The paper has been slightly rewritten to improve its readability. In
particular, we now restrict ourselves to von Neumann entropies. The title and
the abstract have been changed. The references have been updated. 10 page
