The EGT system is driven by thermocouple-type sensors. As such, things things RARELY, IF
When they do, they lie low, not high.
And EGT indication that is above desired/expected is typically a true indication.
So, where do you go from here? Engine health. You say he's cleaned blades... good.
What about, as asked above, pneumatic leaks?
From there, you're really going to look at the FCU, but only if there is an associated error in rotational speed of the N2
Not knowing which type of engine is on the L-39 (if he can't figure this problem out, he's more than welcome to donate the airplane to me... I promise I'll take good care of it), I can't get into specifics because I simply don't know them.
But, generally speaking, a high EGT indication is a TRUE indication, and is telling you that turbine blades, compressor blades, or FCU are having health problems or other issues.
And the FCU is a stretch, at that; the only thing an FCU (hydromechanical, which I'm assuming we're dealing with here, not digital) is sensing is RPM... not pressures, not temps, not what you had for breakfast, JUST RPM. So, unless there is also an RPM issue, stay out of the FCU.
He does realize that EGT rises continually as the engine ages... right? If it's gone past a limit, then... he's looking at a three-bolt fix, as we used to say on JT