On Oct 30, 2022, publishing to a custom domain on FlutterFlow was announced via a newsletter:

To start you can publish one custom domain per Pro or Teams account, additional options coming soon!

Looking at the Pricing, they encourage you to pay annually (25% discount). Also, the Pro and Teams tiers are the most expensive, at about $50/mo. That's a bit much just for that one feature that I want to use. But oh well. They do save me a lot of time, and it's worth that much.

The doc hasn't been updated yet.

The doc still says that custom domains are not supported yet.

Going into App Settings, Web Publishing, I input my custom domain.

The UI then tells me to set up two records on my registrar. An A record and a CNAME record. Once I update my domain, I clicked "Connect" on FlutterFlow, and it worked. It also showed me that I get one free connected domain:

How much will other custom domains cost?

I really hope further custom domains don't cost too much.

Cheaper workaround

One other way to publish to a custom domain would be to get the Standard subscription ($25/mo), download the code, compile it to Web myself and deploy to Firebase Hosting. However, the Standard subscription does not have Git integration. Only Code download. That means it might be a bit of a manual task to download the code, then compile and deploy.