Chicken drumsticks should be cooked at 400 degrees in a preheated oven according to Better Homes and Gardens. They should be cooked for 35 to 40 minutes, and the meat should not be pink when done. A meat thermometer can be used to ensure food safety; it will reach 180 degrees Fahrenheit when the drumsticks are thoroughly cooked.
Chicken drumsticks can also be cooked at 375 degrees, but this will increase the cooking time to 45 to 50 minutes. Chicken should be placed with the skin facing up in a shallow roasting or baking pan. It can be seasoned first if desired, with olive oil and salt and pepper or other spice rub.