With an item like hot sauce, storage can get tricky. While it will typically store fine at room temperature, there are a few details to consider with each specific product.
Hot sauce contains vinegar, salt, and chilies, which all prevent bacterial growth to some degree. If vinegar is the first ingredient on the label, no need to worry about storage.
However, if other ingredients are more prominent — or if there are any fruit additions — then make sure that you refrigerate so you don’t risk bacterial infection.
It's also helpful to consider the amount of sugar, which acts as a preservative. Regardless, it never hurts to put hot sauce in the fridge, since it extends shelf life.