Super weird Yaris 2003 A/C problem

[zoharby]

I have a 2003 manual Toyota Yaris with a very strange A/C problem, which got me and the A/C shop baffled.
History:
1. Since I bought the car 8 years ago, it had a weird problem where the A/C compressor would not start working when starting the car after it had been parked in the sun for a few hours. Only after some 5 minutes it would suddenly kick in and start cooling.
2. Recently, it also started cooling poorly (mostly on low RPMs), and the shop said the compressor is dead and needs replacing.

So, they replaced it with a brand new compressor. Both problems are now gone, but instead an even weirder and more annoying problem came up:
* Whenever I reach a speed of ~50 Km/h and go over ~2500 RPM, the compressor would disengage and stop cooling completely.
* This is not a problem with the TPS sensor, as releasing the gas pedal at high speed would not cause it start cooling again
* If I shift out of gear at high speed and let the RPM fall to idle, it will start cooling even if still at high speed
* I thought it might be a high coolant pressure problem, but just hitting the gas while parking in neutral does NOT cause the problem to appear. It will cool perfectly at neutral on 2500-3500 RPM. This only happens when the car is above some speed.

Any ideas? I can't even think what could cause this weird problem...