We have primary and backup internet providers, but our UAG is only accessible on the primary IP address and hostname. So if our primary internet goes down, we can't access Horizon due to public DNS pointing to the primary IP address.
I think I need to set up a second public DNS entry, (for example vdi2.example.com) with a different IP address. Is it possible to add another interface to my existing UAG's with a second hostname, or do I need to create a completely new UAG with its own IP and hostname "vdi2.example.com"?
You will probably have a listener on your primary internet firewall that is forwarding traffic to the UAG (or the load balancer in front of the UAGs). Setup a second listener with a separate IP address on your backup internet connection. It should also forward traffic to the UAG (or UAG LB).
You've then got a choice of how to give access, depending on what works best for you:
Note that you might need to watch for Origin Checking as the new DNS FQDN might not match what the Connection Servers are expecting. The load balanced name used to initiate a connection would not match the actual Connection Server name. This can cause the Connection Server to reject the request. See the Origin checking section of https://techzone.vmware.com/resource/horizon-configuration#post-installation-configuration