Yeah, wrong question. I could go on trying to define this. There is no relation as the question is asked. There is an answer if you ask the question of a specific turbine.
The goal of the manufacturer is to produce the most thrust possible, to do so he would want to be as efficient as possible. Using heat and converting it to thrust is the goal so a more efficient engine has higher thrust and lower exaust temperatures.
A fine example of this is during a start, egt climbs with engine speed and epr. Thrust is measured by N1 or EPR. at a certain point the EGT starts to "rollback" or decrease as engine speed and thrust continue to climb to idle. This is the only time we can see efficiency increase.
As we advance thrust to T/O settings EGT climbs, different temperatures and air densities effect egt and thrust.