When I turn the HVAC unit on, it effectively cools the house. After a few hours, however, I notice that the fan in the outside unit has stopped running. The air handler continues to operate, but there is no cooling anymore. If I turn off the system, wait an hour and start it, the outside fan starts operating again and cooling is effective once more. I seem to notice that when it is hotter (afternoon), the outside fan operates for a shorter time before stopping.

Sounds like the fan is going into an "overheat mode" and that is why your seeing the short cycle times. Some units offer a thermo protection circuit. If the sensor is malfunctioning then it can caus the same thing. Either way you need to check it.