That said, if you make either correctly, there shouldn't be anything in there that goes bad. I keep my (homemade) browned butter/ghee on the counter, along with my coconut oil.
My ghee doesn't set up, by the way. I don't know why. It's very liquid -- literally "butter oil" -- but tastes delicious. I drizzled some on a sweet potato last week and it was so good I almost cried.