2000 Jeep Cherokee Starter

Engine Performance problem
2000 Jeep Cherokee 6 cyl Four Wheel Drive Automatic

My daughter has a 2000 Grand Cherokee. The other day she could not start her vehicle. Her boy friends father rapped on the started and it started. She did replace the battery since it had lost most of its cranking power when she brougt it to the store for testing. Since she replace the battery, the vehicle has started, and two initial times during the day, when she turned the key, nothing happened then, it started. This morning it was about 12 degrees, the vehicle started right up. Could it be starter was frozen up and replaceing the battery did the trick? Or should I attempt to just replace the starter?

Thank you