From French's website
Q: Do FRENCH'S® Mustards need to be refrigerated?
A: Dijon & Horseradish Mustard will lose heat and distinct flavor, so we encourage refrigeration for both. For all other mustards, refrigerating will help maintain its flavor; however, it is not necessary if you prefer your mustard to be room temperature. There are no ingredients in mustard that spoil. "Refrigerate After Opening" is not required for food safety. It is recommended to maintain optimal product flavor.