What is the name of the oath taken by its utterer, "By God, I have cooled down from it," when he did not intend that, but rather meant that cooling the device is good? And does it entail expiation?

If you falsely swore that you caught a cold from the air conditioner, this is a "yameen ghamoos" (a false oath sworn deliberately on a matter of the past or present), and it is a grave sin. The majority of scholars hold that there is no expiation for it, but rather one must repent to Allah Almighty. However, Imam Al-Shafi'i and some scholars believe that expiation is obligatory along with repentance, based on the verse: "Allah will not impose blame upon you for what is unintentional in your oaths, but He will impose blame upon you for what you intended of oaths" (Quran 5:89). Combining repentance and expiation is safer. However, if the words merely slipped from your tongue unintentionally, then you bear the sin of lying without the sin of the oath, and there is no expiation for it. It is advisable to safeguard one's oaths.