Does Ubuntu approve an application which is written using Adobe AIR technology? Yes, we know Adobe abandoned support for AIR for Linux. But still we can use Adobe AIR 2.6 and create applications using that.
Looking at the Canonical partner repository, it seems that Adobe AIR is only offered as a package available from there from Ubuntu 10.04 to Ubuntu 11.04.
So starting from Ubuntu 11.10 Adobe AIR is no longer supported as far as I can tell.
As an alternative, you might want to look at another technology to write your application. Also, I've noticed that there is this non-open-source but gratis application that is written in AIR and bundle their own Adobe AIR package (I guess from Adobe). Have a look at their .run file for Linux to see how they install the app.
In any case, this is just an example for your reference. If Adobe is abandoning Linux support for AIR, you should really consider using another supported technology if Ubuntu is one of your target platforms.