  Danchin and He (Math. Ann. 64: 1-38, 2016) recently established the global
existence in critical $L^p$-type regularity framework for the $N$-dimensional
$(N\geq 3)$ non-isentropic compressible Navier-Stokes equations. The purpose of
this paper is to further investigate the large time behavior of solutions
constructed by them. More precisely, we prove that if the initial data at the
low frequencies additionally belong to some Besov space
$\dot{B}_{2,\infty}^{-\sigma_1}$ with $\sigma_1\in (2-N/2, 2N/p-N/2]$, then the
$\dot{B}_{p,1}^s$ norm of the critical global solutions exhibits the optimal
decay $(1+t)^{-\frac{N}{2}(\frac{1}{2}-\frac{1}{p})-\frac{s+\sigma_1}{2}}$ for
suitable $p$ and $s$. The main tool we use is the pure energy argument without
the spectral analysis, which enables us to \emph{remove the smallness
assumption} of initial data at the low-frequency.
