Whether an air conditioner will run when it's colder outside your apartment than inside depends on the type of AC system you have and how it's set up.
Traditional Air Conditioner: Most traditional air conditioners, like window units or central AC systems, are designed to cool the indoor air by transferring heat from inside to outside. If it's colder outside than inside, the AC's thermostat may detect that the desired temperature has been reached and shut off the cooling operation. However, some older models may not have sophisticated thermostats and could continue running even if it's colder outside.
Heat Pump: Heat pumps are a type of air conditioner that can work in reverse during colder temperatures. In addition to cooling, they can also provide heating by extracting heat from the outdoor air and transferring it inside. Therefore, in the case of a heat pump, it can continue running and provide heating to the indoor space even if it's colder outside.
Dual-Mode or Multi-Mode AC Systems: Some modern AC systems have dual-mode or multi-mode capabilities, combining traditional air conditioning with heat pump functionality. In this case, the AC system can switch between cooling and heating modes, depending on the indoor and outdoor temperatures and the desired temperature setting.
To sum up, if you have a traditional air conditioner and it's colder outside than inside, it may stop running once the indoor temperature reaches the setpoint. On the other hand, if you have a heat pump or a dual-mode AC system, it may continue running to provide heating to the indoor space, extracting heat from the outside air even when it's cold.
Keep in mind that the specific behavior of your AC system will depend on its design and thermostat settings. If you have concerns about your AC's operation or energy efficiency, it's a good idea to consult the user manual or contact a professional HVAC technician for advice.