PA-28 pilot performing an engine break in flight reported the aircraft burned more fuel than planned and they ran out of fuel.

Narrative

Had to finish an engine break in procedure where the aircraft was operated at 2600 RPM and mixture full rich. Planned fuel consumption was 12 GPH for a 3.5 hour flight. I had 48 gallons usable giving me 30 minutes of reserve at destination; ZZZ. Fuel ran completely empty approximately 4 miles north of ZZZ. I was at approximately 5500 ft. I was able to glide in to Runway X with no damage to the aircraft; property; or myself. It was later discovered that the aircraft was burning a much higher than expected amount of fuel at 15 GPH causing me to run out of fuel before reaching my destination. I did make contact with ZZZ Approach on XXX.X; I believe; and advise them of my situation. I did call them after landing safely on Runway X at ZZZ. At approximately 30 minutes from the destination I switched from my left tank; which the engine was still running; to my right tank. The fuel gauge did show about 0 gallons remaining in the left tank at the time. My right tank was still showing around 15 gallons in my right tank so I continued to the destination assuming I still had enough fuel to make it. The right fuel tank gauge then started to quickly move towards 0 and the engine died. I switched tanks back to left where the engine did start back up but only for about one minute. I then started to make calls to ZZZ and flew an engine out procedure. When time permitted I made contact with Approach. It was on later; shorter; flights that the 15 GPH fuel burn was discovered.