Bringing this up again from a previous archived post:
Can we or can we expect to use a custom domain name with the Oracle cloud service?
The statement "At this point in time, you cannot use a custom domain name for your APEX application" is true from a pure Oracle point of view. However, if you have a really hard requirement that this is implemented there are always ways to get this done in a more "interesting" and "creative" way.
If you build your application pure with Oracle domain names, as you normally would do, and you want to have this translated to custom domain names you could trick the system by implementing a reverse proxy somewhere else that has the name you desire. You can configure you reverse proxy to communicate with your APEX application and your users would communicate with the reverse proxy. However, you will have to write some rules with MOD_REWRITE and such. However, you see this being done in other situations so, even though I have not tested it and it is most likely not supported by Oracle, this could very well work.