I have a bunch of hot sauces. 13 different kinds at last count but they come and go so I may have more now or less. Most likely more because the homemade are coming on line.
I keep any open bottle in the fridge. None say "Refrigerate After Opening" but I do because I think it keeps them fresher longer.
It's not always about the heat but the taste. I haven't noticed any real loss of heat but on the shelf the taste can change faster then if refrigerated.
I do let them come up to room temp before using them.
Don't know if that makes a difference or not?
No scientific evidence. Just an observation from someone with an asbestos tongue and a cast iron stomach.