Refrigerating Hot Sauce is not required and hot sauce will not go bad if it's not refrigerated.
You can keep the hot sauce out at room temperature for as long as 3 years although it loses it's freshness after being opened.
You can keep the hot sauce fresher by keeping the hot sauce refrigerated but it's not required to keep the hot sauce from spoiling.
I've kept hot sauce opened up with the cap on the bottle for 3 years before and it was still very safe to use and eat.
It just doesn't taste as fresh as it would if just opened.
So no hot sauce does not go bad if not refrigerated.