Lethargy, a general low energy level and lack of response, can signal a serious illness in any cat or dog. In a diabetic, it can be a signal of dangerously high or low blood sugar, or of other health problems. Cats are lone hunters by nature and therefore cannot afford to be lethargic -- the appearance of this symptom in cats indicates that the problem is serious.
Lethargy is one of the symptoms of uncontrolled/poorly controlled canine or feline diabetes. Untreated or not adequately treated, the diabetic's tissues are unable to use glucose, amino acids and fatty acids properly[1]. It normally resolves quickly after starting insulin therapy.
Unusual lethargy for unknown reasons might be a veterinary emergency.