Yeah, my cat Sulo - the Jekyll&Hyde cat you may recall - does that sometimes. He used to sleep next to me and he'd just get up in the middle of the night and stare at... nothing.
Weird thing is sometimes I could feel something myself, too, like a cat or a dog jumping on the bed, moving about, then lying down next to my legs. Sometimes I'd feel it so clearly that I had to turn in my bed and have a look whether it was Sulo. I like to sometimes imagine that it's the spirit of Osku, a welsh corgi pembroke who used to live in our house before we moved in and who I personally knew very well (I used to walk him and take him to agility, match shows and such). Sadly he got run over by a car when he was a only something like three years old and died.
Anyhow, it's more likely to be just the sensation of the blood in my veins and heart thumping and all those things that I seem to notice much more at night when everything's so calm and quiet, combined with a vivid imagination. I've also felt it when I've been away from home, in another country, so the Osku-theory probably doesn't hold (unless his spirit is following me around!). And it's stopped since I got my new dog, a rough collie named Elli (hence the nickname I use on this site).
So, I don't really know what cats are thinking of when they do that... but I do kind of like the idea of seeing spirits - even though it might creep me out in the middle of the night!