I have registered with Uptime Robot. You can have it monitor a URL. It does this by sending a HEAD request every 5 to 10 minutes and checking for an OK response. This request triggers the fastcgi cache with an empty content. If I understand the wiki correctly I can't not cache HEAD requests (http://wiki.nginx.org/HttpFcgiModule#fastcgi_cache_methods).
The only (dubious) fix this I can see at the moment is to add an IP exception to the nginx config. However that effectively leaves the site open to abuse by someone being malicious.
Is there a way I can just cache GET requests? Or requests with body content size > 0? Or is there a better way?