...the GTE has a FCU. How will that affect the idle speed?
Depends COMPLETELY on the design of the FCU. Some will govern to a constant speed, modulating fuel as required to achieve that. (i.e. colder day, more fuel).
And some will intentionally reduce rpm to maintain constant idle thrust, though not necessarily the same rpm as in the first question.
I have worked on both types.
Incidentally - on a engine for supersonic use, at high Mach (high inlet temp) the idle speed is likely to be increased a great deal - up to 100% maybe. This is to insure that the aircraft inlet keeps "swallowing" all the air rammed into it.