The temperature of the stove pipe should run as hot and safely as you can.
A good temperature to keep the stove pipe and stove pipe chimney flu is around 250 F any temperature lower than 250 F you risk building up creosote.
If your flu temp starts getting to 500 F or above then you're likely running too hot and overfiring the stove.
I would recommend getting a wood stove pipe thermometer that has a magnet on it to mount to the stove pipe to show you the temperature of the stove pipe.