Here is the situation:
I own a Google Apps for Business account.
I have two domain names registered :
It doesn't seem possible ATM to bind an App Engine application on a Google Apps Secondary Domain (sounds like a bug).
But it works fine on Google Apps Domain Alias.
That should work.
As of now, naked domains ( example.com ) are not supported by App Engine.
Instead you can make a Web Forward from example.com to www.example.com. This can be achieved in various ways depending on where you've registered your domain and if the registrar offers DNS services.
If not, you can use Zoneedit:
These changes might take some time to propagate
On the google apps page you had to add thee www web-address on the applications settings tab (what in my understanding you already did)
Hope this helps, and if not you, then maybe others :). + zoneedit is a great & free tool for DNS management, and you can also define custom mail forwards and stuff :)
(Ps. the screen shots are taken from a working app engine example site that runs from www.secondhand-extra.ro )
I think the real answer is to be found in Google's Limitations for multiple domains:
"Custom URLs for accessing Google Apps are available only for the primary domain. The Admin console enables the administrator to set a custom URL for various services. The custom URL applies only for users accessing the services at the primary domain. The same limitation applies to custom web address mapping for a Google Site."
It's quite a severe limitation, since it requires you to open a new Google Apps account for each custom domain you want to register, even if all domains belong to the same organisation.
At my goog app manage domain front page, I go to domain settings -> add domain alias for www.zzz-zzz-zzz.com. So I added www.zzz-zzz-zzz.org
At my domain registrar, I declare the following redirects:
zzz-zzz-zzz.com -> www.zzz-zzz-zzz.com zzz-zzz-zzz.org -> www.zzz-zzz-zzz.org
After that the following urls would all go to the same place:
google app engine does not allow mapping naked domain. you can use a redirect from myapp.com to www.myapp.com.
"Due to recent changes, App Engine no longer supports mapping your app to a naked domain. If your domain registrar supports URL redirects, you can redirect from http://yourdomain.com to e.g http://www.yourdomain.com or http://appid.yourdomain.com."
http://code.google.com/appengine/kb/general.html#naked_domain