This is actually a fairly common issue :) Our app is directly connected to the Facebook app and it verifies the account details directly through both the Facebook app and iOS Settings. Sometimes when Facebook details are changed, this process needs to be refreshed.
Please try to completely close the 8 Ball Pool app, log out of the Facebook app and log back in. You can also log out and back on through your iOS Settings (Settings > Facebook) to make sure this is completely refreshed.
Once this is done, please try to connect to Facebook again through the 8 Ball Pool app :) let us know how it goes!