A difficulty for this line of thought is that space is expanding (and so cooling). You can imagine a static space with no contents, but we exist in a universe where the thermodynamics is actually wired in.Consder, if we had a volume of space that was completely empty, a perfect vacuum, then would time pass within that space?
If there was nothing there to change and, therefore no change happened, could we say that any time passed?
It is true that at the heat death in a flatly balanced universe, there would be very little change - as little as possible change. But there would still be change. And the speed of light would still be the yardstick on this change.