I have another vehicle that checks out fine it seems, but happens to get off to a slow start cooling. Like you really need to get in it and drive it to get cooling. Then it is ice cold. Then sometimes on a long drive at highway speeds it will get somewhat warm occasionally. Not all the time. It's been doing it for a few years and nothing has cropped up. I suspect a pressure switch issue or something with it's fancy climate control crap.
Liberty... never had that issue. It's cold cold at idle, at speed, etc. It was cold until it didn't work at all. I never really did get a true diagnosis but it would run then quit. It was either the clutch, or the non-replacable inline filter got plugged and caused restriction causing it to trip out on pressure. Nobody really diagnosed it, just quoted me a massive bill to replace it all.
I ended up just ordering all the parts online and replaced it all myself. After taking it apart I can say the compressor itself did not fail. But the clutch could have been going out or the inline filter could have been plugged. I had all the new parts in case the compressor had failed. So I just put them all on. It has worked great but the aftermarket line does make a humming noise or I still have air in it somehow. As good as it works, I just recommend backflushing the original line to make sure the filter is not clogged or source a new oem line if you have to replace parts later. I don't think it's air.
Anyways, I'd take it in somewhere reputable, have the charge checked while it is hot, and have them check the clutch out.
I am going to say in your case, maybe you are low on charge, and once the RPM gets up and the air flow increases it cycles off on pressure, then comes back on. Either that or the clutch is weak and the RPM causes it to come and go. But that is easy to test in the driveway. Put it on, rev to same RPM and let someone watch to see if it kicks off.
You might also test both modes at highway speed... recycle on and off. Could be an air blend situation and the speed. Mine seems to blow and cool better now with recycle off now that I think of it, and I don't know if that used to be the case. Its been years since the whole family has went on a trip in it, where as my wife would turn on the recycle to reduce outside smells from coming in. So I don't remember. Worth looking into though.
