It's the clear language of the U.S. Constitution (Article I, Section 3) related to impeachment that has nothing to do with the Fifth Amendment prohibition against multiple prosecutions for the same crime.
It was Bshaw who asked why it wasn’t a double jeopardy argument. Whether you call it ‘double jeopardy,’ or something else, the question before the Supreme Court isn’t whether a former President has immunity from prosecution because he was impeached and found not guilty, it is whether a former President can be tried for actions taken while in office.