I think that if William wants to live with his girlfriend, he should be allowed. Especially since he apparently sought permission and approval from his father and grandmother.
King or not, it's the 21st century and many couples live together before marriage in this day and age. Some couples end up getting married whiel others do not. But living together enables couples to know if they are compatible and can deal with the day to day annoyances that are bound to crop up when you have to get used to someone else's habits and quirks.
At the height of Diana's popularity there was a lot of praise for how compassionate she was and how she would hold people's hands and really talk to them without gloves or whatever. While the Queen and Princess Anne and Prince Charles were staid, cold and unapproachable individuals. There was great demand for the monarchy to change with the times and to take Diana's lead. Slowly the monarchy did evolve and since Diana's death we've seen the Queen and Charles's approach to their royal engagements soften and become more approachable.
So shouldn't the same standard be held for William? Shouldn't the monarchy change with the times now too and allow William's life to reflect that of thousands of other people his age who also live with their girlfriends and boyfriends prior to marriage? Living with Kate would not be any different than what Johnny Smith down the street is doing. William has a right to a personal life too, even if he will one day be king. And in that personal life he deserves to be happy and fully committed to the woman who will be his wife.
Charles wasn't fully committed to the (first) woman who became his wife and look at where that's gotten us.
I applaud Charles and Elizabeth for allowing William to live with his girlfriend. They are putting his happiness before the "family business" unlike with Charles, when they put the family business before Charles's happiness, putting a virgial aristocrat in his life rather than his true love.