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The fermentation process generates heat from within the fermenter so it should be a bit warmer than the ambient temp. I would get some ice or wrap a wet towel around it...but like others have said, it's not ideal but it'll still be beer.

Alrighty then, Question.... Say I get some ice in a towel and wrap up the fermenter... would the temp changing from time to time effect the end result?

It would if you could not keep it at a steady temp, sometimes slightly high temps are better than fluctuating ones