In hot weather RPM increase significantly

Thanks!
The starting RPM is normal from hour parking and if engine is not hot. If engine hot after half hour driving the idle RPM is high. When engine warms up the RPM is not increase, only engine became hot the RPM was increasing. And the fan is working when engine is hot even A/C stopped. And if start A/C the fan Will start moving. Is it indicating the sensor okay? I don’t know how to test temperature sensor. I will test the sensor.L will try one by one following your instruction.

How long have you owned this vehicle?
I bought this new Montana in 2003.

Have you had it in hot weather before to know this is not normal behavior of the vehicle?
Yes. Since last year.

When you are running the engine and it gets hot and the RPM's come up, what does the temperature gauge say the temperature is? Where is the needle?
The needle at one memory above middle. It is about 100+80+4= 184.
I checked coolant radiator and all pipes were hot, but reservoir tank is not hot, and the pipe line connected to the tank was warm but not hot after run 20 miles on the highway. When l run on the highway 70 mph the RPM was 2,000 RPM, but after run 20 mile when exit to local free way at 40 mph, the RPM was 1,500, and kept the speed until stopped at home. If run on the highway at this state, to reach 70 MPH the necessary RPM increased to 2,500.
But water temperature gauge memory kept at middle without change no matter the engine got hot or not.
Many thanks to you!

Thanks!
Great instruction and analysis! Your view are very reasonable. It looks like this case is really complicated situation. I will check one by one.

I think there is no TPS/TPA issue, because in the start and under half hour driving RPM is normal. But after that point the RPM increase a lot, and coolant temperature increased a lot, and speed is very hard to increased even foot above 2,000. If there is possibility of coolant sensor bad and thermostat is bad? If there is possibility when temperature increase vacuum hoses leak that caused the problems?
Thanks for your instruction and comment.

I checked transmission fluid level was okay, and clean. Recently I found a hole on the Air Cleaner Intake-PCV Valve Tube Hose 24508188, and replaced with new one, and replaced all 6 spark plugs. And running was okay without misfire. But old symptom of “hot-nutral-RPM surge high” still cause problem. Later I found thermometer increased near to the warning line after 20 minutes driving on the highway. Then, I suspected thermostat and replaced it, and had to disconnected one vacuum hose which connected to back of the engine, and throttle body. After restore all parts, when I start engine there was error code P0300-multiple misfire Detected. First time and car was vibrated abnormally, and engine service soon light turned on. After clear the error code when I start again the RPM surged from 400 to 3,000 rpm's. I start again and drove car out of garage, then there was high sounding like siren from two different sound scream source. I stopped soon. I found there was one more new error code added : P0131: oxygen sensor, circuit low voltage, bank1 sensor 1.
I think high surged RPM with high oxygen content caused oxygen sensor malfunction.
Now with two code P01300 and P0131 I do not know how to work on these issue. Thanks a lot for your expert review, instruction and help!

Thanks a lot for your instruction and helps! I will do my best follow your instructions. I also suspect my improper thermostat replacement procedure may caused high p0300.
Radiator pressure too high problem:

1. I could not refill original volume of released coolant from top of radiator.
2. Radiator pressure is too high that coolant spurred out a lot when when l started engine short time.
3. Before the p0300 appears radiator pressure was high that coolant leak found from breeder screw base.
4. Two kind of High sounds may be from breeder screw. I tried breeding air at the same time and opened two breeder screw.

If improper Thermostat replacement and throttle body restore may cause coolant system pressure too high?
Thank!

You were right that "coolant issue may be a cause", and 'thermostat may be okay". My radiator pressure has been high, and I did not know why, but recently found radiator cap may be caused it that the exceeded high pressure could not emit through cap to tank. I ordered new cap. A new p0131 code appeared. I will replace Bank 1 O2 Sensor 1, but I do not know if I need to replace other sensor down stream that located on the exhaust pipe upstream of muffler. Any way, I will reset spark plugs, thermostat, throttle body, etc. Let me know your precious comment and instruction on my current situation. Thanks a lot!

I reinstalled thermostat and throttle body, and cleared the two codes. But car not start up. So I guess starter was bad and disconnect it. I forgot disconnect negative end of battery first and there was a spark with positive and negative mixing during
starter replacement. I disconnected battery ends and changed starter. I connected positive end of battery cable first, and then connecting negative end of cable, but the starter cracking without engine key on ignition). Is it possible current mixing that changed ignition switch position to switch on? How to check and fix it? Thank you very much for your instruction and helps!

Thanks! Okay. I already stopped starter cracking without ignition key turn on when battery cable connected.