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One is not allowed to tell a non-Jew to do a melacha on shabbas, but one can hint and say "it is cold in here", hoping he will realize and shut the air conditioner.
Is one allowed to explain to the non-Jew that he (the Jew) is not allowed to say straight out to do a melacha, so that, from this explanation, the non-Jew will better understand what to do than if he were only told "it is cold in here"?
The Mishnah Brurah 307:76 explains as follows (with a bit of spice from me). There are two types of Issuei Melacha of Amira LeNochri. Well actually there are really three but the third isnt really about Amira Lenochri. The third one is the Issur to speak on Shabbos about Dvraim Ha'asurim to do on Shabbos. This applies even if you say I will do this Melacha on Sunday. The speech itself is the Chilul. Because this is the Issur, any type of hinting is OK because hinting is considered thought rather then speech and it is speech that is Ossur, not thought. (Although the Gemoro says MeMidas Chassidus is not to even think about doing Melachas after Shabbos).
The next Issur is to request a non-Jew do a Melacha for you on Shabbos regardless if you benefited from it in the end. It is simply forbidden to request a non-Jew to do a Melacha for you. Therefor the Rema says certain types of hinting are also included in this Ban. This is because if you wink at the non-Jew to do it and nod your head at the air-con fo example, that is considered a request. Speech it isnt, and that is why it is not included in the first Issur, but a request it still is. On that the Mishnah Brurah says that is only by a hint in a tone of command, ie a nod, or put the lid on the cap when you want to refer to a totally different Melacha which the non-Jew understands. Because these are essentially requests, just not spoken, these are Ossur.
However to say that it is not light enough in here, or I cannot put on the light, because you are not requesting at all, however obvious it may be that is your intention, it is not considered a request and therefore one may ask the non-Jew. (Obviusly it must be in a way where there is no benefit from the Melacha of a non-Jew that was done for a Jew on Shabbos, because even when you are allowed to ask a non-Jew to do it, you may not benefit. One way that is consiedered not benefiting, is if he put on a candle for you and there was alraedy some light there so it is not considered a Ha'anoh)!
Now for my own hypothesis
I see no reason why it should be Ossur to tell a non-Jew that you are not allowed to ask him to do it on Shabbos. You are not saying it as a request, it is a hint and therefore is in my humble opinion Muttar! (A harder question is to tell the non-Jew during the week, 'why didn't you do that thing last week', this is a debatable topic and is disputed amongst the Rishonim)
Just to add another point, in the specific case you mentioned, there are authorities who permit asking the non Jew directly to turn off the air conditioning, as according to them turning off the air conditioning is only rabbinically prohibited, and one may ask a non Jew in case of need to do something which is only rabbinically prohibited.
There are two prohibitions of Amira L'Akum.
The first problem can be avoided through hinting, e.g. "It's cold in here, and we may not turn of the air conditioning." Note that the halachos of what is considered hinting vs. saying is fairly complicated.
The second problem is not an issue if it's not considered a true benefit (such as turning of a light to allow for easier sleeping). In the case of it being cold, however, providing heat is a real benefit, so we have a problem.
That being said, there may be room to permit it in this case. Since "everyone is sick when facing the cold", it may be possible to be lenient, depending on whether the activity is prohibited min haTorah (Biblically) or miderabanan (Rabbinically), how cold it is, whether there's somewhere else to go to avoid the cold, and whether there are young children suffering.
