I was having the "unable to connect" problem and I've noticed that sometimes when Windows loses its Internet connection, it will sometimes tell other apps that the site is down until the page is loaded in a browser. I loaded us.gran-turismo.com in Chrome and it was able to load after that.
Try loading http://www.gran-turismo.com/, http://us.gran-turismo.com/, http://eu.gran-turismo.com/, and http://store.playstation.com/ in your browser to try to get Windows to see that they're online.
If server is down the widget stops working and won't be able to work until you go home and sign back in to the ps3 and restart the server. I guess I found that out today.