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.
Lethargy is one of the symptoms of uncontrolled/poorly controlled canine diabetes. Untreated or not adequately treated, the diabetic's tissues are unable to use glucose, amino acids and fatty acids properly.  It normally resolves quickly after starting insulin therapy.
Unusual lethargy for unknown reasons might be a veterinary emergency.