What can happen during extreme temps, either too hot or too cold, is the actual drug portion of the medication can break down or looses efficacy. Often the carrying agent that a medication may be in, ie: a drug that is in a oily carrying agent, such as an injectable vitamin or antibiotic, may not freeze, but the extreme temps can still harm the actual drug in the oily liquid.
Same with ointments too. Some drugs are affected more than others and some not at all….just depends on the drug or medication.
Some are not affected at all though.
They really should be kept in a temp controlled area such as a house, but in a boarding situation sometimes that is not possible. Can the BO provide a controlled environment? How about taking them back and fourth with you when you go there?
Trying to build a “cool” box is a good idea in theory, but unless it’s next to a cool area such as the cold water pipes or and air conditioner, there isn’t a whole lot that can be done to keep a box cold without ice packs, as I‘m guessing the temps don't go down enough over night to cool the box.