When setting up an application gateway (Standard V2), I was asked to create an IP address (used Standard SKU for the IP). Is this IP address static or dynamic? I'd like to create a CNAME record and point my custom domain to this address, but I wasn't sure if the address would change without my consent.
This is explicitly covered in the Application Gateway documentation, specifically within the Application gateway components article:
Static versus dynamic public IP address
The Azure Application Gateway V2 SKU can be configured to support either both static internal IP address and static public IP address, or only static public IP address. It cannot be configured to support only static internal IP address.
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The DNS name associated with an application gateway doesn't change over the lifecycle of the gateway. As a result, you should use a CNAME alias and point it to the DNS address of the application gateway.
TL;DR - The static/dynamic nature of the assigned public IP can be configured within the V2 SKU however best suits your use case.