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It's not the problem with unity.
It the expected behavior,you have to download the targeted sdk version even if you build with Android Studio.
That weird. It should work without downloading anything.
Do you by chance clear your Android SDK?
Another problem you should watch out is Android Studio update. When you update an Android Studio there is a bug (don't know how to reproduce it) which set the SDK path empty string.
Since the Android SDK path is empty, the Android Studio may download the SDK path.
Advice : Android Studio update are rarely stable so only update when you have to.