The purpose of a sump pump is to prevent flooding of the basement.
If the basement remains dry, then the pump is doing its job.
If the house waste line were frozen, collapsed or otherwise obstructed, then drains throughout the house would be backing up, including the toilets.
This leaves the float switch, which is causing the sump pump to run when there is nothing for it to pump.
A one horse sump pump is huge and would never run for half an hour straight unless the float or the switch is bad.