The phrase "The Rubik's Cube" sounds odd because you can't normally use an article with a pre-nominal genitive if the pre-nominal itself wouldn't normally take an article.
You can say "the paper", "the professor" and "the professor's paper". You can say "David's paper" but not "*the David's paper". (Although note that if talking about the sculpture "the David", you can say things like "the David's left hand". And, because of Donald Trump, you could say "the Donald's hair".)
You can't say "the Rubik" and so "the Rubik's cube" seems ungrammatical if you think about its component parts.
What's happening is, of course, that "Rubik's" isn't acting as a genitive anymore but rather "Rubik's Cube" has been reanalyzed as an opaque compound noun. It's just still written in terms of its components.