What I have learned is that when you stop an EC2 instance, the next time you start it up the instance will likely have a new IP address and thus a new public DNS name provided by Amazon. If you have a CNAME pointing to Amazon's public DNS, you will need to edit the CNAME so that it points to the new public DNS.
I did not want to edit my CNAME every time I restarted the EC2 instance, so I use an Elastic IP. With an Elastic IP, every time you associate that IP to an instance Amazon recreates the same public DNS, which is what I have the CNAME pointing to. Consequently, whenever I want to restart my EC2 instance I do the following:
1. Launch the instance in EC2
2. Associate my elastic IP to that instance, then wait for a few minutes for the public DNS to propagate
3. Connect to my instance via RDP using the CNAME and verify that the OPML editor is running
From that point forward I can then log in to Radio and my River refreshes. Of course, all of this is moot if you never stop or terminate the instance.