It works on Chrome, but somehow it doesn't work on Firefox..
You can play a YouTube video at twice its speed on Chrome because you're using an HTML5
<video> player. It's almost always automatic on Chrome because Chrome seems to support everything the player needs:
HTMLVideoElement, H.264, WebM VP8, Media, Source Extensions, MSE & H.264, MSE & WebM VP9.
FireFox, on the other hand, only supports the first half:
Hence the player is rarely an HTML5 video object.
However, you can turn it on manually. Go to http://www.youtube.com/html5, and click on the blue "Request the HTML5 player" button:
You can later click "Use default player" if you have any problems. However, you should now be able to change the speed of most of your videos like in Chrome: