Your own domain
Use an address you already own.
After you keep an app you can give it a name of your choosing (family-books.uarpieapps.com instead of quiet-harbor-4471.uarpieapps.com) and, on Personal and above, attach a domain you own. Apps made before apps had their own domain keep their .uarpie.app address, and a new name for one of them stays on .uarpie.app.
Adding a domain
From the app's Address tab, or ask your AI ("use books.example.com for the family app"). You get two DNS records to add where you bought the domain: one points the domain at your app, the other proves it is yours so a certificate can be issued. Once they resolve, the certificate is issued automatically and renewed for as long as the domain stays attached.
On Free, attaching your own domain is refused with a note saying it needs a paid plan. A domain already attached keeps working if the account later moves to Free.
While it is pending
Status shows Waiting for DNS. It usually takes minutes, sometimes an hour. The app's own address keeps working throughout.