In Normal mode, Max Climb is obtained at a set TLA. The FADEC calculates the Max TO (or outside the takeoff envelope, Max Con) - which is also a set TLA (roughly 2 degrees from the forward stop - from there to the forward stop all you'll get is the 'headroom').
N1 increases linearly between the Max Climb TLA and the Max TO/Con TLA. Below the Max Climb TLA, the TLA/N1 relationship follows a shape that roughly gives linear thrust (the thrust/N1 relationship is very non-linear at lower N1s).
So if you set Max Climb - by definition the TLA won't change as you climb since that's what the FADEC controls to.
I don't recall what's done on the GE90, but on some engines above 30k, Max Climb and Max Con merge - so you end up with a big flat above the Climb TLA setting.