What theories exist connecting volcano temperatures to eruption probabilities? Wikipedia names thermal monitoring as one of the methods for predicting volcanic eruptions. I was able to find some papers (First, Second) on such thermometry and while they are challenging (I retired from physics years ago), I will consult them for the methods.

What I completely fail to find information on, is how current thermal behavior is supposed to be linked to future eruptions (other than: "it’s getting hotter, that could be magma rising upwards"). I realize we don’t have complete theories, but I hope you can tell me what kinds of ansatzes there are.
I have not heard of complete theories connecting volcano temperatures to eruption possibilities either. If you mean subsurface magma temperatures, it is uncommon for there to be direct observations / readings for a particular volcano. Fumarolic temperatures, on the other hand, are directly indicative of current volcanic activity and are much more easily measured (example). In addition, geochemical analysis of fumarolic gases can be related to eruption probability (example, example).

Even in models of magma ascent, temperature differences are not the main focus, but rather pressure differences are (example). This is directly related to g.z.'s comment that volatile content certainly can be a dominant eruptive trigger.