  We investigate the $S^3$ free energy of $\mathcal N=3$ Chern-Simons-matter
quiver gauge theories with gauge group $U(N)^r~(r\geq2)$ where the sum of
Chern-Simons levels does not vanish, beyond the leading order in the large-$N$
expansion. We take two different approaches to explore the sub-leading
structures of the free energy. First we evaluate the matrix integral for the
partition function in the 't~Hooft limit using a saddle point approximation.
Second we use an ideal Fermi-gas model to compute the same partition function,
but in the limit of fixed Chern-Simons levels. The resulting expressions for
the free energy $F=-\log Z$ are consistent with each other at the leading and
first sub-leading order. The Fermi-gas approach also hints at a universal
$\frac{1}{6}\log N$ correction to the free energy. Since the quiver gauge
theories we consider are dual to massive Type IIA theory, we expect our results
to match sub-leading corrections to the holographic dual free energy, which
have not yet been fully investigated.
