Recently I have been very interested in the concept of a thermal time. The basic premise is that unidirectional time coordinate we experience may not be a fundamental property of some GUT (grand unified theory) scale physics but is instead an emergent phenomenon. At first this sounds somewhat anarchical as physical laws mostly focus on predicting the state of a system at some future time and almost invariably involve some kind of description of the behaviors of a system under dynamical evolution. The idea is at once fascinating but also unconventional enough to be something you need a really good reason for doing. If you are genuinely curious try reading Rovelli and Connes .
The story begins with the Tolman Ehrenfest effect. Static observers who measure Temperature in gravitational fields find that it depends on the gravitational potential at the point where the measurement is made so that
is a constant (interestingly this scaling is usually associated with the relation between the coordinate time and the proper time measured by a clock traveling along a world line) [2,3]. In other words a long column of fluid in thermal equilibrium will naturally have a temperature gradient with hotter liquid at the bottom. The difference is tiny; nevertheless it implies general relativity is deeply intertwined with the temperature of your beer tower.
What if at some level statistical physics and thermodynamics are fundamental and what we perceive as time is an emergent property. We can postulate that the universe is described by a generally covariant theory which treats all coordinate directions evenhandedly and time is an artifact of the thermal state that we are immersed in.
There is no place for a preferred time variable in general relativity; furthermore if we look at the physically observable quantity measured by a clock it is in fact the proper time along a world line. The coordinate time which parameterizes the field variables,
because the equations of motion are manifestly Lorentz covariant it really doesn’t matter which you choose. They are all just reparameterizations.
Intriguingly thermal states break Lorentz invariance. They pick out a preferred coordinate system; in fact a thermal bath preferences the Lorentz frame in which it is at rest. This thermal state may be used to define a preferred physical time. Yet the background theory remains generally covariant.
In thermal equilibrium our thermal states can be described by Gibbs states; these encode information about the Hamiltonian and the dynamical properties of the system are attributed to this thermal state rather than direct Hamiltonian evolution. To some extent it is valid to assume you are always working in a thermal regime as you stereotypically don’t have access to the full microscopic state of a system and hence you simply reconstruct the microscopic state from macroscopic observations
 Rovelli and Connes [arXiv: 9406019v1]
 Rovelli and Smerlak [arXiv:1005.2985v5]
Tim-Torben Paetz [#://www.theorie.physik.uni-goettingen.de/forschung/qft/theses/dipl/Paetz.pdf]