Does the imaginary time path integral for the partition function imply that temperature set a characteristic time scale for quantum systems?

In the ordinary path integral, the action is an integration over the time your interested in. In quantum statistical mechanics the integration is over an imaginary time with the limit $\frac{\beta}{\hbar}$. Does this imply this is the longest time relevant to the system is $\frac{\beta}{\hbar}$

I'm motivated to think about this because I'm trying to understand the analytic continuation used to go from imaginary time green functions to the retarded real time greens function used to study response. My reasoning is vaguely this:

If you only have information about the response function on the imaginary axis at discrete frequency intervals, then this lack of information would some how show up in the analytic continuation to the real axis.

The answer to your question is no: there is no relation between the integration limit of the path integral in imaginary time (${\beta}$), and a time scale relevant for the study of the system. Imaginary time is called time but has no physical relation to time, as considered in the time evolution of the system.