When a swf is embedded in a page and is not cached locally in the browser, two requests for the swf are fired by the browser. Here is a simple example.
The first request returns an expected 200 OK, and the second request triggers a 304 Not Modified after the first request has completed, indicating the file is now cached locally.
Why is this 2nd request fired and how can I prevent it?
I just verified that this happens for me, as well.
At first, I thought this was due to a bug with Firefox. See FAQ question #16 on this page. The bugs they reference appear to be fixed, however.
So then I looked at some of my own web sites that embed SWFs using SWFObject 2.2. The double-GET did not occur. I think the difference is that in my code, I'm calling swfobject.embedSWF in the <head> of the page instead of the body. Putting the script in the head is what the SWFObject 2.2 documentation recommends (this is a bit of a departure from older versions of SWFObject).
So here's what I'd try if I were you:
I think I figured it out. I read the bug reports for Firefox more carefully, and this looks to be caused by AdBlock Plus, which I have installed. First, I verified that the bug still occurs with your updated page, and reproduced it a second time. Then, I disabled AdBlock Plus and tried the same test. The double request was no-longer happening. So if you have that extension installed, or perhaps a similar extension, that may be the cause.
This is the bug report about the plugin being double-instantiated, and this is the bug report for the fact that AdBlock Plus causes it.
I believe this can also be triggered by running Firebug - try checking your browser requests using LiveHTTPHeaders (or some other http sniffing app like Charles) while you have Firebug disabled.