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I've had to do the 24 hour urine a couple times. One was when I was pregnant and I had to keep it in the fridge but didn't have to be super careful about chilling it. The other was for histamine and you have to be super careful for that one, chill each sample before adding it to the jug and make sure it's cold while you're taking it to the lab and that they put it in the fridge as soon as you give it to them.
A cooler in the bathroom is a good idea, just add ice as needed. I have a couple coolers I bought at Walmart for $20 that will keep ice for 24 hours.