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I've seen Does an expanding universe cool down?

So I understand an expanding universe cools down, my question is, does an expanding universe reach maximum entropy faster than a non-expanding universe?

I ask because heat transfer is faster the larger the difference in temperature. What I don't understand at this point: is expanded space colder than the already existing space, causing energy to spread out faster?

Or am I off base and the speed of light limits the speed at which maximum entropy can be reached.

scm
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