In discussions of Yang-Mills instantons it is often stated that one should sum in the path integral over all contributions of fluctuations around all the topologically distinct vacua labelled by winding number $n$.
Usually there follows a discussion on $\theta$-vacua, which are basically a linear combination of $n$-vacua, in the sense that $\lvert \theta \rangle = \sum e^{in\theta} \lvert n \rangle$ and that the procedure of summing over all fluctuations around instanton contributions in the path integral could be replaced by applying the usual rules to a Lagrangian where the term $ \sim \theta (*F, F)$ is added, where $(.,.)$ denotes the Cartan inner product.
I don't understand how this can be. Doing this, one seems to replace the path integral as a sum over contributions from different $n$-vacua by just the path integral expanded in a unique $\theta$-vacuum. Indeed, this seems to come down to summing over a particular linear combination of $n$-vacua. But WHY should you only take into account only a particular $\theta$-vacuum, even though the $\theta$-vacua are inequivalent (they have different energy density)?
Why should e.g. QCD not just be studied in the true vacuum (the one with the lowest energy density) which is just $\theta=0$? Is this because, the $\theta$-vacua, too are topologically protected?