No, things don't heat up from sunlight alone in orbit. The sun is giving a mere 1.4 kW / m2, so the BG would aborb the same amount. The energy dumped into a black hole would have to be vented through flickering (the cooling ability of a starship must be on par with its powerplant power so this should be no problem. If the BG was on at all times the heat buildup would eventually be troublesome, but not nearly as much as the heat buildup from the powerplant itself. The heat from the PP cannot leave the BG when it is on either, right?
IMTU BGs work like perfect blackbody radiators and will heat up when irradiated and have a temperature. When ships try to overload a BG they can measure their success by the IR temp of the BG, hot enough and the BG will explode.
IMTU BGs work like perfect blackbody radiators and will heat up when irradiated and have a temperature. When ships try to overload a BG they can measure their success by the IR temp of the BG, hot enough and the BG will explode.