Entropy is a state of energy change. From higher energy to lower energy. That rate of energy can simply be offset by cooling, slowing the molecular motion and therefore rate of energy loss. Or by acceleration and adding energy offsetting the energy lost by the molecular motion. Or by a gravitational field adding energy offsetting the energy lost by the molecular motion.
Entropy is just a word for energy loss. Energy is unavoidable in the discussion of time or entropy....
Which brings us back to distance... which is really what is being measured....
The term "entropy" was originally coined by engineers during the development of the steam engine. When they did an energy balance around the early steam engines, there was always missing energy.
The energy inputs and the outputs were never exactly the same, based on all the traditional energy assumptions and measurements. Entropy was the term used to account for the missing energy, so they could close the energy balance.
The second law says that the entropy of the universes increases with time. This means that the net energy, absorbed by and tied up within entropy, is not reversible, and is therefore lost to the universe, in terms of being reusable energy. We can reverse entropy, on a smaller scale, but it takes more energy than we get back, causing net useable energy to be lost by the universe. Entropy absorbs energy and takes this energy off-line in terms of a universal energy balance; 2nd law.
Entropy and time work in similar ways. Time does not repeat or recycle itself, like reusable energy. Once time is expressed, it is lost forever. The past might be recorded in various structural changes; mountains, but the past can't be reproduced from these recorded structures. The past, like the energy within entropy, is preserved, but lost to the present and future, since it is not useable.
Entropy is also a state variable, which means for any given state of matter, there is a fixed amount of entropy contained in that specific state. Entropy values are not random, but are constant for any given state. For example, the entropy of water at 25 ◦C and 1 atm is 188.8 joules/(mole K). The entropy of that state is always the same, no matter how we get there, or who measures it. Entropy is implicit of how matter is organized; facade of matter. Time is perceived as the facade of matter changes, and the unusable energy within entropy changes the facade.
Say we had two references, one reference (A) is where space-time is contracted and time moves slow. In the second reference (B) space-time is expanded and time moves fast. I place identical factories in each reference. Both factors make widgets and both have the exact same production rate. Each factory also makes defects in the widgets, at the rate of 1 per hour. Since a defect changes the facade of the widget, a defect reflects a change in entropy; different state of widget. We will throw away or recycle the defective widgets since we can't undo the defect by reversing time. All we can do is alter it in the present; recycle.
If I stood in a third reference, which has space-time between the two, and I looked at the two factories, side-by-side, we notice that because time is moving faster in factory B, its production rate of widgets is much faster with respect to our reference. Since time in factory A, appears slower from our reference, its production rate of widgets is much slower.
We also notice that the increase in entropy, as reflected by the defects, is moving faster in factory B. This also means that energy is becoming unusable at a faster rate, in factory B. When time moves faster, the rate of lost universal energy; nonreversible, increases. With the energy lost, we do not have the energy needed to reverse time, so it can only move forward.
The expanding universe; space-time expanding, is using up reversible energy at a faster rate; red shift. This energy being tied up in an increasing variety of universal facade. A highly contracted universe; singularity, where time moves slow, has very little facade variety. Expansion and the speeding of time, allows for variety in exchange for unusable energy.