Hi, @Brad Mathews Thank you for reaching out to us with your query about Azure Postgres. I'd be happy to help you with this.
There is no stop option for Azure Postgres single server and this is by design. The best option is to take a backup and delete the server or another option is to scale the instance down to a lower pricing tier. This can help you save costs while keeping the instance around for a bit.
Regards
Geetha