If there is a power outage on Shabbat, it is permitted to turn air conditioning back on that was shut down due to the outage.
This ruling has been issued by the Tzomet Institute, an Israeli rabbinical think tank that addresses halachic (Torah law) questions surrounding technological issues.
The ruling is in keeping with the principle that Shabbat (Sabbath) prohibitions may be violated where saving lives is at stake. Turning on electrical appliances is one such prohibition but it may be violated during a potentially lethal heat wave.