  Most of the computational evidence for the Bose$\small-$Fermi duality of
fundamental fields coupled to $U(N)$ Chern$\small-$Simons theories originates
in the large-$N$ calculations performed in the light-cone gauge. In this paper,
we use another gauge, the 'temporal' gauge, to evaluate the finite temperature
partition function of $U(N)$ coupled regular and critical fermions on
$\mathbb{R}^2$ at large $N$. We first set up the finite temperature gap
equations, and then use tricks explored in arXiv:1410.0558 to solve these
equations and evaluate the partition function. Our final results are in perfect
agreement with earlier light-cone gauge results. The success of our 'temporal'
gauge calculation potentially opens a path to computations that are awkward in
light-cone gauge but more natural in the 'temporal' gauge, e.g. the evaluation
of the thermal free energy on a finite-sized sphere.
