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Hi John,
thanks for explaining you reason for sticking with temperature.Its a good one. Also temperature and heat are often used interchangeably in everyday language. Turn up the temperature or turn up the heat, not a lot of difference because they go hand in hand. You are right temperature is very significant for biological organisms. As you know, the enzymes that control metabolism are heat sensitive. They can be ineffective at very low temperatures and become damaged and less efficient at high temperatures, giving a range between where there is optimal performance. Even rates of chemical reaction that are not influenced by enzymes or catalysts are affected by heat.
However if rate of change was giving passage of time then hot things would be disappearing into the future as time is passing quicker for them and cold things would be being left in the past. Subjectively it might seem for the hot entity that time is passing quicker- as it is more active, getting more done and ageing faster compared to a cold thing. That is slow and sluggish, doing little. As its metabolism is slower, growth rate and the damage occurring due to metabolic processes and accidental damage, wear and tear would also be slower. (If they are higher organisms they could have an internal biological clock that is set by fluctuating light levels and that will give the organism some perception of passage of time that might differ from external reality, especially of it has been kept under unnatural conditions.) Despite their different individual experiences they still exist at the same time and externally time is passing the same for both.
It seems to me heat /temperature/kinetic energy is correlated with -how much- spatial change is occurring -simultaneously- (especially at the atomic scale), it could be a little or a lot, but universal minimisation of potential energy which is the default change that will always occur, at all scales, that also occurs simultaneously (each little change being the kinetic energy minus potential energy) is correlated with passage of time. Comparing rates confuses matters because a faster rate is not faster passage of time but it is faster alteration of arrangement within the same passage of time. So its not to my mind kinetic energy that is driving passage of time.
Going back to the box of balls analogy from long ago. There could be two boxes one representing the hot entity and the other the cold entity. Both are stationary. In the hot box the balls are moving around a lot and in the cold one the balls only move a little. Simultaneously both boxes could be moved to a new spatial position representing the motion of the Earth (as the entities are from their perspective stationary). That movement if done gently will not affect the rates of the processes within and the hypothetical entities represented would not be aware that it has occurred. That seems a bit like passage of time to me.
Not sure you'll be convinced. I think I have said something similar previously.