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3rd party YouTube app Metrotube is running a closed beta to test Chromecast support coming to their Windows Phone app.
They are also working on adding support for casting to your TV and Xbox, suggesting Miracast is also being explored.
MetroTube is a full featured YouTube app which includes features such as continuous playback, which will play all videos in a playlist without intervention, and an audio mode, which will let you play the audio of a video only.
The current version costs $0.99 US, but the free trial is fully functional and does not expire.