The only thing that cooking oil will supplement is your pet’s intake of fats. Extra fat derived calories are not useful for coat or skin quality. Fats are essential to a pet’s diet, but there are many types of fat and only specific ones will do. Natural oils are secreted by glands in the skin and protect hair and skin, composed of protein, from drying. Cooking oil is not a source of nutrition and is unnecessary.
Concerned about your pet’s appearance? Speak with a professional. If your pet is found to be healthy and has a poor coat despite adequate nutrient, prepared supplements of the specific essential fatty acids might well benefit your pet. A healthy pet does not require any additional fat.