I just discovered that a brand new tube of my prescription toothpaste decided to sneakily hide under my electric frying pan–which I then used to fry an egg.
So?
What is the ingredient prescribed? Normal sodium fluoride is not going to break down or anything.
If the tube is plastic I’d be a bit concerned about it partly melting into the toothpaste. But not enough to not use up the tube.
Unless it sprung a leak; fighting with a leaky tube for the next 3 months is Totally. Not. Worth. It. Ask this cheap-ass how he knows.
I can’t imagine the paste would be harmed by being heated.
I’m trying to figure out how a tube of toothpaste would get under an electric frying pan!
I’m not the OP. But …
I unpack all my groceries on the counter in the kitchen. Don’t you? On a crowded counter I can certainly imagine one small item getting jostled unseen against a small countertop appliance like an electric fry pan. Which is one of the few appliances with enough airspace underneath to hide something.
You’re probably not going to jostle your toothpaste under your e.g. coffee maker or toaster. But under something up on legs? Sure.
Don’t forget that while fluoride is a key help, you get benefit from toothpaste beyond just that. The abrasives and the act of brushing alone provide a benefit when used properly towards the cleaning of the teeth.
As for other effects - fluoride, whitening, breath freshening - I’m sure there could be SOME effectiveness lost to the heat, but doubtful all. What I’d probably do though is set the heated tube aside as my backup, buy a new tube, and go on. And if that day comes when I finish a tube but forgot to buy new, the backup is there.
But I normally buy whatever is cheapest/on-sale at the store, so being out $4 for a medium sized tube isn’t a tragic loss, just a frustrating one.
Depends on the type of toothpaste. If it’s efficacy is based on abrasives in the paste it will work fine.
The OP specified “prescription” toothpaste. Last time I had some of that stuff it was $25/tube. So maybe a bit less disposable than e.g. Crest depending on the OP’s finances or natural cheapness. And also raises the possibility it has more complex ingredients that might be heat sensitive.
My prescription toothpaste simply had more flouride than the OTC stuff, so would be (probably no more or less heat sensitive than Crest. But without knowing specifically which toothpaste we’re all talking about, even a pharmacist would be hard-pressed to provide an FQ answer. Not that that has stopped me or anyone else from offering IMHOs. 
Yes, that is what mine is: a generic with 1.1% sodium fluoride.
How hot does it get under the electric frying pan? I’d think the bottom would be well-insulated so you don’t scorch your counters.
Many have little or no insulation. They are raised high enough for air flow underneath to keep the counters from getting very hot at all. The toothpaste tube is at least half that distance between the pan and the counter so it would get fairly warm. I’ve pulled utensils out from under counter top pans and they can get uncomfortable to hold.
My counters are too cluttered for that, so I tend to unpack my groceries on my stovetop.
Now there’s an overheating surprise in the making. I admit I sometimes do that too when the counters are really overstuffed. But I always cringe while I do it.
All my microwave ovens (I’m on my third one now, I think) have little legs so there is a bit of airspace below. That didn’t stop Oven #1 from scorching the countertop. I now have a sheet of 1/4" plywood (well, particleboard actually) underneath my microwave oven.