In a medium saucepan cook onion and garlic in hot oil over medium heat until onion is tender, stirring occasionally. Stir in apple juice, tomato paste, vinegar, brown sugar, molasses, paprika, horseradish, Worcestershire sauce, salt, and pepper. Bring to boiling; reduce heat. Simmer, uncovered, about 30 minutes or until sauce reaches desired consistency, stirring occasionally.
Brush sauce over beef, pork, or poultry during the last 10 minutes of grilling. If desired, reheat and serve additional sauce on the side.