In Arnold's Mathematical Methods of Classical Mechanics, he leaves as an exercise to show that if $S(E)$ is the area enclosed by a closed phase curve of energy $E$, then $T:=S'(E)$ is the period of a particle travelling along. But $S$ is just the entropy of a collection of microcanonical ensembles with that Hamiltonian, and $T$ would be therefore an inverse temperature.
This made me wonder if there's a deeper connection here. Could it be that to every equlibrium statistical system there is a corresponding periodic phenomenon, of which the temperature is a kind of frequency? My first thought is that in equilibrium the various degrees of freedom oscillate about their mean values and that maybe temperature is a measure of the frequency of that oscillation. But I don't think that's the case, because in Arnold's example $T$ corresponds to the biggest curve in the manifold $H=E$ (which happens to be unique because it is one dimensional) and the oscillations are certainly not the "biggest" such curves in the statistical case (even forgetting about how to define "biggest curve" in a manifold with larger dimension than $1$).
My second thought was that I have seen, even if invoked in a rather mysterious way, the use of the so called imaginary or Euclidean time which turns the Maxwell distribution into a periodic function, of period $\frac{\hbar}{k} T$ ($T$ being the "period" defined before, not the temperature). This seems to be a periodic phenomenon of period $\sim T$; however like this answer points out, such a phenomenon is useless physically because it corresponds to imaginary energy. This is furthermore unsatisfactory because in Arnold's example the period is of an actual phenomenon that can take place in phase space, unlike the periodicity of the Euclidean circle.
Is there a deeper connection happening here or am I grasping at straws because of a funny coincidence?
EDIT: As pointed out by Tobias, $S$ is not actually the entropy but, denoting entropy by $\Sigma$, $e^{\Sigma /k}$. Nonetheless, T would then be $e^{\Sigma /k}\Sigma '(E)/k$, that is, $T$ would still be (proportional to) an inverse temperature.
