Also, does EGT vary according to altitude at similar N1 speeds?
This is a fairly complex performance question but it's related to "corrected N1".
In brief - as TAT falls with altitude, the corrected N1 increases, even at the same indicated N1. This means the engine is working harder, relative to its capability at that altitude. Working harder = higher EGT (of course
Tom355uk, you, & I know it's really ITT)