Google has made it known several times that they will not be supporting the RT and have no plans to do so in the future. This is a bit rough from the stand point of the RT users however from their point of view there is little adoption of the RT.
"OK, so let’s start with the bad news. Chromecast devices require Chrome to be installed for full functionality and you just can’t install that on a Surface RT or Surface 2.
However, all is not lost as there are a few apps in the Windows Store that will let you use a Surface RT or Surface 2 with a Chromecast. AV Cast and Tube for Chromecast are a couple of examples. I can’t comment on how well they will work for you but, they are available."