[David L.]

No. There's no implication of actual 'rushing' to do something ie haste. Both marching and double time are regulation ways that troops move in formation. Obviously, double time will get you there quicker! Marching is formal, double time, 'informa'l but still army regulation.
The implication in your example, is that four minutes of jogging (double time) is more energetic/tiring than 4 minutes of marching. The sergeant has come into the barracks to get the men out of bed and doing marching and push-ups before breakfast. He made it double time as a kind of punishment for Hume's slowness in getting in line with the other troops.