Thank you for posting in the answers forum! That's a great question. When you change your router or modem the Slingbox will not automatically connect to the new one. You will first, need to hard reset the Slingbox to factory settings. After you do this and have a solid power and network light; you will need to go online to setup.slingbox.com and setup the Slingbox again.
Below is a link that tells you what to do when getting a new router:
Hope this helps!
The Sling Moderation Team