I have just tried this myself and found that the temperature data from the API matches the temperature data on the local “/json” page. Since you are using “/json?live=true”, the readings are not averaged every two minutes, so there could have been an event that increased the temperature on your device that the API would not show yet.
Is it possible that you are already applying a correction to the data through the API?
We apply an eight degree correction to temperature readings, but that is only on the PurpleAir map. A bit more information is found here: PurpleAir Sensors Functional Overview.