I don't think he actually does care at all. The man is a sociopath who uses people and discards them the moment they cease to be useful. He played up his "love" for Catelyn because it hides his true motivation, which is obtaining power. His house is one that's poor and of small standing, so if he married a Tully, he could potentially become lord of Riverrun and move up from there. Ned's brother and then Ned himself fucked that up.

With his recent raise up to Lord of Harrenhal, he has the standing to marry Lysa Arryn, Cat's sister, and become Lord of the Vale.

By "saving" Sansa, he controls what people believe is the only surviving Stark. He intends to leverage that to become Lord of Winterfell.

I won't spoil anything, but he's a fucking brilliant schemer who has no problems manipulating people to make things go in his favour. Let's just say he's one of the primary reasons the War of the Five Kings happened and he seems to be the only one reaping the benefits from it.