I don't think that is true- I've heard that things like nursing and wearing certain types of after pregnancy support belts can help shrink it down faster, but I'm not sure that a cold beverage would do anything to cause your stomach to not be flat after birth.
Good luck to you with baby #6!
It is hard to see how drinking anything cold would make a difference to the muscle tone of your stomach. Were you told this by someone with medical training? If not by a doctor, have you checked this with a doctor?