Inside air temperature has nothing to do with the AC clutch being turned on or off. The climate control always turns the AC on (unless you press the AC button to turn if off). It always cools, once the interior temperature reaches your setting, it starts adding heat to avoid going too low. It does not turn off the clutch to the AC. (Also, "60" is a manual setting that will cool at maximum even if the temperature in the car is less than 60.)
The AC clutch is turned off if:
1. Engine temperature is too high.
2. Engine RPM is too low (under 600 RPM)
3. The gas pedal is floored.
4. AC low side pressure is too low.
5. AC high side pressure is too high.
6. Under hood temperature is under 40 F (this is a function of the low pressure being too low due to the temperature).
7. Evaporator air discharge temperature too low.