ok, when we come to turbines i will give here some impressions on the turboprop GA aircraft i currently fly- a cheyenne III with the pt6a-41 engines. the pt6a-41 is a turboprop two shaft engine which incorporates a three stage axial compressor followed by a 1 stage centrifugal compressor driven by a 1 stage compressor turbine. after this we have a 2 stage power turbine which drives the prop via a gearbox. the both shafts are not mechanically coupled and the low pressure turbine drives only the prop - so we call the pt6a is a free turbine.
the engine is like common nowaday design flat rated - so the rated power output is a mechanical limit and the engine is able to keep rated power above ISA or keep rated power in thinner air when you climb until its thermodynamical limit ( ITT or compressor speed) is reached. in other words- the engine could develop on ground more power that the gearbox is approved for.
basicly on this engine you give with the power levers an input to the FCU ( fuel control unit ) to set a target compressor speed. in regard to air density and outside temperature a given amount of fuel is needed to keep this speed. this will result in a given force to the power turbine and a given torque - so power output.
at take off you are mostly torque ( so power output) limited and the turbine is at its mechanical limit . the ITT and compressor speed are below its limit. when you climb out and do not touch the power levers the compressor speed stays the same. the ITT also but torque and fuel flow decreases. this is due the fact the FCU ( fuel control unit) keeps like said a target compressor speed . in a climb out the air gets thinner and the "resistance" on the compressor stages also. so the compressors try to spin faster and the FCU has to decrease fuel flow to keep the same speed. due to less fuel and gas driving the power turbine the torque ans power output also decreases.
when you want to keep the same power output in a thinner air in a climb you will have to push the power levers more and more forward. this will result in a faster and faster spinning compressors and a higher and higher ITT until at a given altitude you match the maximum ITT or compressor speed. here the turbine reaches its thermodynamical limit.
sooo... when the air gets thinner and the compressors deal with a lower pressure ratio ( in pistons compression) the ITT rises .thats a fact. i found and attached a pic at our top of climb in FL 280 with a cheyenne III with pt6a-41 engines so you can have a look what the torque, ITT, compressor speed amd fuel flow is here. at this altitude the engine is at its thermodynamical limit - so the compressor speed is at company limits resulting in a given ITT and torque far below its redline ( mechanical limit)
now we can talk why it is so a a turbine engine.