You can easily map a custom domain to a Convox application by creating a CNAME to your load balancer hostname.
Balancer Hostname
You can find the load balancer hostname(s) for your application using convox services:
$ convox services
SERVICE DOMAIN PORTS
web docs-web-R72RMTP-326048479.us-east-1.elb.amazonaws.com 80
Configuring DNS
Create an appropriate DNS entry to map your desired custom domain to your Convox app. In the example above one might create the following DNS entry: