Temperature as the independent variable of Lagrangian

I was thinking about applications of the Lagrangian and I started to toy with some ideas and tried to come up with interesting twists. Immediately I thought it would be interesting to use temperature as oppose to time as the independent variable.

Essentially, instead of

$ \int f[x, \dot{x}; t]dt $

Something like

$ \int f[x, x'; T]dT $

There are a few issues with this:

1. 
   

   Is this even a legitimate method? Given that we use time and take the derivative with respect to it I'm a tad bit skeptical as to whether this can be legitimately done. What would the derivative mean in this context.

   
   


2. 
   

   The other issue with this is that temperature really is just giving us stochastic information, so how would one be able to create kinetic energy equation that is inherently random? Even if the idea of using temperature is invalid the question about randomness could still be useful for time based Lagrangians.