Air conditioner compressor won't stay kicked in
 

The freon level is at the right pressure but the compressor will stay on for a while and then kicks out. It's awfully hot to have this thing acting up. Is there a sensor or something going bad?

The A/C compressor is not supposed to keep running all the time. It is normal for the A/C compressor to kick in an out every 2 to 3 minutes. If it does kick in and out in a matter of few seconds then this is more likely low freon. Is your A/C not cold enough that you are concern about your compressor?

How long does it take to cycle on-off-on? If it stays off long enough that the air is warming up there are a few things that may be going wrong. There are high and low pressure switches that can go bad, stick, or just plain quite functioning. There is also either an expansion valve or orifice tube that over time can become contaminated and cause A loss performance. There is also an accumulator/drier that can fail do to a buildup of contaminants. And the A/C compressor clutch can go bad from wear. Hope that helps.

Two other possibilities came to mind: There is a cut out/cut in switch that controls the cabin temperature. Those do go bad on occasion but it is rare. Also, if the system is overcharged the high pressure switch will shut the system down as a protection measure and the system will not engage until the pressure lowers to a point below maximum.

What are you using for a gauge to determine that it has enough refrigerant? If it's just a cheap low side gauge with no pressures and temperature markings on it, you're probably not understanding what you're seeing. The temperature bands are pretty worthless, and it doesn't have a fast enough reaction to read the temperature during suction.
My honest guess, you're a can low.

We have noticed over the last 1.5 months, and this very hot summer with a black car, that our a/c is going on and off. Usually for 10 min. at a time at least. It tends to kick back on. It happens anytime of the day. We realize it because the air is warm coming out of the vent, but we also notice that the exterior temperature gauge on the dashboard drops to 22 or 25 degrees Fahrenheit. It will go up and down and then once the a/c kicks back on, it starts to record an accurate outside reading.

What does that tell you about our problem?

Could be a weak clutch magnet or a worn clutch, clutch getting too hot. Refrigerant level may be low. It is difficult to tell without inspecting.