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Can I use 気温 and 温度 when asking for the weather temperature? Both terms indicate temperature. Which one is more commonly used?

Though 温度 is a generic word for temperature, we prefer the specific 気温 "air temperature" everyday when we mention the weather, in conversation or in forecast. Same for 水温 of water, 体温 of body, 室温 of room etc. Especially, it'd almost sound like a joke if you described someone 温度が低い instead of 体温が低い (a languid person??).

In my impression, 温度 is only preferred when it would otherwise make a tautological expression (*食塩水の水温) or the object has no convenient compound in the form of ○温.

温度 is a more general term for temperature, whereas 気温 is the atmospheric temperature, specifically. Expect to see the latter in more technical or scientific sets such as the weather forecast.

気象きしょう庁ちょう : "Japan Meteorological Agency defines「気温」as

通常は地上1.25～2.0mの大気の温度を摂氏（℃）単位で表す。度の単位に丸めるときは十分位を四捨五入するが、０度未満は五捨六入する。

I could not find the original page in the site, but searching the definition of temperature on the site gives this piece of pdf : Chapter 2 Measurement of Temperature. It explains,

2.1 Definition and units

Heat balance difference of atmosphere between regions creates temperature distribution. This temperature distribution generates wind current along with cloud and rainfall phenomena. Thus, atmospheric temperature is one of the most important meteorological elements as well as wind and precipitation. WMO recommends to measure atmospheric temperature at the height from 1.25 to 2m above ground at a representative location of region, as standard.

So, it seems 気象きしょう庁ちょう follows the guideline of World Meteorological Organization: "WMO".

気象きしょう庁ちょう uses the system of「アメダス（AMeDAS）」: "Automated Meteorological Data Acquisition System" to monitor precipitation, wind direction/speed, temperature and sunshine duration, such as the following image :

And, according to 「気温について」, the temperature we are knowing should be

Q. 〇〇時の気温は、〇〇時の瞬間の気温ですか？

A. 〇〇時の前１分間の平均気温です。例えば、「１２時の気温」として表示されている気温は、１１時５９分から１２時００分に観測された気温の平均です。

The temperature has displayed at the time of 〇〇 is measured by the average temperature monitored during last one minute.

So, all in all 「気温きおん」is measured by the system like above.

Things which are difficult to check the temperature by the system such as temperature of oil to fry, checking the temperature of water in your bath is described by 温度おんど .

室温しつおん, which is used in setting the temperature of air conditioner, is bit subtle. By definition, it can be a part of 気温きおん according to this site. However it is saying "indoor temperature". Whereas, 気温きおん is normally used to describe "outside temperature". So, it should be separable.