What makes the difference is the mass. ...[T]he earth is a big mass and it slows down time. If you go to a bigger mass, like Jupiter, it's stronger. … - Carlo Rovelli

Whatever the statistical state ρ is, there exists always a variable t_ρ, measured by the thermal clock, with respect to which the system is in equilibrium and physics is the same as in the conventional nonrelativistic statistical case!
This observation leads us to the following hypothesis.

If we want to understand the past we must do so on its own terms, and disregard the future of that past, but if we want to understand the present we better not disregard the past steps that were essential for getting to the present. This is of importance especially for those of us engaged in trying to push ahead the scientific path of discovery today. We are not much interested in what scientists did wrong, there is too much of that. We are interested in what they did right, because we are trying to copy them in this, not in that.

The notion of time familiar to us may... be reconstructed in special physical situations, or within an approximation, as... the "surface of a liquid" disappears ...[at] the atomic level, or "temperature" ...makes sense only in certain physical situations and when there are enough degrees of freedom.