  The $\mathbb{C}P^{N-1}$ sigma model at finite temperature is studied using
lattice Monte Carlo simulations on $S_{s}^{1} \times S_{\tau}^{1}$ with radii
$L_{s}$ and $L_{\tau}$, respectively, where the ratio of the circumferences is
taken to be sufficiently large ($L_{s}/L_{\tau} \gg 1$) to simulate the model
on $\mathbb{R} \times S^1$. We show that the expectation value of the Polyakov
loop undergoes a deconfinement crossover as $L_{\tau}$ is decreased, where the
peak of the associated susceptibility gets sharper for larger $N$. We find that
the global PSU($N$)=SU($N$)$/{\mathbb Z}_{N}$ symmetry remains unbroken at
"quantum" and "classical" levels for the small and large $L_{\tau}$,
respectively: in the small $L_\tau$ region for finite $N$, the order parameter
fluctuates extensively with its expectation value consistent with zero after
taking an ensemble average, while in the large $L_\tau$ region the order
parameter remains small with little fluctuations. We also calculate the thermal
entropy and find that the degrees of freedom in the small $L_{\tau}$ regime are
consistent with $N-1$ free complex scalar fields, thereby indicating a good
agreement with the prediction from the large-$N$ study for small $L_{\tau}$.
