Depending on your soap formula, hot & even warm lye solution can cause the soap to overhet very quickly. Some soaps will crack as they set up in the mold when they overheat, but not all do; again, it depends on the formula & your environmental conditions as well, but more on the formula.
A soap recipe high in CO cracks easily. I have stopped the top from cracking from advancing by simply lifting the old up off the surface upon which it sat and placting it on a slotted rack which provides more air flow beneath the mold, and help cool it down faster. In that instance it was a hot day in Texas and the recipe was about 30% CO (not my recipe; I won't use soap with that much CO). When making 100% CO soap for laundry soap, even that doesn't stop it from cracking, but it doesn't matter because it's not for bathing anyway.
So the simple answer, is, Yes, temperatures do matter. But so does the formula. I prefer to soap with oils that are clear and not cloudy (the hard oils that are melted to liquid), which is usually fairly warm depending on which oils I use in a particular recipe, and the lye solution at room temperature or a bit warmer if my house is cold (like it is in the winter).
With a high liquid oil containing recipe, I can soap much cooler so in that case, I tend to do everything at room temperature.
And it's not only temperatures that can cause soap to heat up, but some fragrances and some additives & additional ingredients can as well. So that's why we say' it depends....'