... and that the engine is on temperature control.
Not quite, Buster15. The engine is not on "temperature control" but is being governed to maintain the N2 speed called for by the power lever angle (not "temperature control", but "N2 control"). This is what the Pilot Training Manual says, and is consistent with every other twin spool engine I am familiar with.
The FCU then needs to increase fuel flow to restore temperate to the damanded figure ...
Half correct. The FCU is increasing fuel flow to bring the N2 back to "on speed" - not the temperature.
The Pilot Training Manual, logic and experience all say the N2 should stay the same (with everything else changing), but that is not what happens. The N1 stays the same and the N2 increases.
???