| This seems to be going off into the weeds. I think what you might be trying to argue is that in vacuum spacetime with a horizon there is a Gibbons-Hawking temperature T_{GH}. A Eulerian observer with a purely timelike worldline could measure T_{GH} at various points along its worldline. In de Sitter space the temperature is proportional to the Hubble parameter H. Unfortunately, there is afaik no consistent or complete microphysical description of the Gibbons-Hawking effect. Additionally, it's so small that the presence of a relic matter field (like the cosmic microwave background) completely swamps it by many orders of magnitude (for H_0 it's 30 orders). So T_{GH}, purely general-relativistic, does not seem like a useful arrow of time in our universe until the latest of epochs maybe. Look instead to the Boltzmann entropy of the matter; we can simply set a low-entropy boundary condition somewhere in spacetime, and then we do not even need to have an expanding universe to have higher entropy in spacetime at increasing distance from that boundary. Alternatively, we could go through some hoops to try to remove the boundary, as Hartle & Hawking among others. Carroll has a list of relevant slides at https://www.slideshare.net/seanmcarroll/what-we-dont-know-ab... None of this has much if any contact with relativistic thermodynamics. In cosmology we are almost always deliberately considering a family of privileged observers and ignoring the others (even though they exist). Our defined cosmological observers see matter (incl. radiation and any horizon radiation) as homogeneous and isotropic. Any expansion or contraction of space is an adiabatic process. \Lambda is nearly zero, but still sufficient to generate enormous spacetime curvature (the purely timelike worldlines of our observers are not parallel). Finally, calculating in the cosmological frame is a convenience, nothing more. |