Police were called to reports of a fire in the early hours of Monday morning at the property in Kentish Town in north London, the area that Starmer represents in parliament.
The man was arrested in the early hours of Tuesday on suspicion of arson with intent to endanger life in connection with the fire and two further incidents, police said.
Police are investigating whether a fire at the entrance of a property in nearby Islington on Sunday and a vehicle fire in Kentish Town on Thursday are linked to the incident on Monday.
Starmer lived in the terraced house on a back street with his wife and two children before he moved into Number 10 Downing Street when he became prime minister last July.
His spokesperson thanked the emergency services for their work on Monday.
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