I use a cooled camera (an ASI1600MM) with the cooling off to image the Sun with short exposures in the daytime and with the cooling on to image the deep sky with long exposures at night. Apart from the extra weight of the Peltier, heatsink and fan there are no disadvantages that I am aware of. Regarding David’s point about frosting up when the cooling is on, none of the more recent ZWO cameras that I own have suffered from this. It was a common problem many years ago but modern cameras are better designed with smaller, well sealed, chambers and, sometimes, window heaters.
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