12. $f(x+T)=f(x)$, i.e., $\cos ^{3}(x+T)-\sin ^{2}(x+T)=\cos ^{3} x-\sin ^{2} x$. By setting $x=0$, we get $\cos ^{3} T-\sin ^{2} T$ $=1,1 \geqslant \cos ^{3} T=1+\sin ^{2} T \geqslant 1, \cos ^{3} T=1, \cos T=1, T=2 k \pi(k \in \mathbf{Z})$. Therefore, the possible values of the positive period of the function $f(x)$ are: $2 \pi, 4 \pi, 6 \pi, \cdots$. Upon verification, $2 \pi$ is the positive period of $f(x)$, and it is also its smallest positive period.
12. The smallest positive period of the function $f(x)=\cos ^{3} x-\sin ^{2} x$ is