I can't even play it. Please give the URL of the web-page with the provided link.

Every time I try to play it I get a URL with a different timestamp (key). Timestamps expire and thus the recording can be prevented.

Although if you can play an HTTP .flv video stream, you can get the .flv file from the cache folder of your browser as described in the following FAQ:
How to record/download/capture/save Macromedia Flash .FLV video HTTP stream from embedded player