  We study the local and (bipartite) entanglement R\'enyi entropies of the free
Fermi gas in multi-dimensional Euclidean space $\mathbb{R}^d$ in thermal
equilibrium. We prove positivity of the entanglement entropies with R\'enyi
index $\gamma\leq1$ for all temperatures $T>0$. Furthermore, for general
$\gamma>0$ we establish the asymptotics of the entropies for large $T$ and
large scaling parameter $\alpha>0$ for two different regimes $-$ for fixed
chemical potential $\mu\in\mathbb{R}$ and also for fixed particle density
$\rho>0$. In particular, we thereby provide the last remaining building block
for a complete proof of our low- and high-temperature results presented (for
$\gamma=1$) in J. Phys. A: Math. Theor. $\textbf{49}$, 30LT04 (2016)
[Corrigendum: $\textbf{50}$, 129501 (2017)], but being supported there only by
the basic proof ideas.
