In a heavy, 1-quart nonreactive saucepan combine the apple juice, sugar and vinegar. Cook over medium heat until the sugar is dissolved. Add all peppers and bring to a boil over high heat. Reduce the heat and simmer for 10 minutes, stirring occasionally. Once the mixture has simmered, stir together the cornstarch and cool water to make a thin paste and use a metal whisk to whisk the paste into the sauce. Continue simmering until the sauce is noticeably thicker, about five minutes more. Remove the sauce from the heat and let it sit at room temperature for 30 minutes before serving.