You can turn AW into a cosmic horror game without many additional rules, I think, if you play up the hugeness of the disaster and the insignificance of the survivors. You might ban some of the more hopeful character types like the Savvyhead and Touchstone.
Change the flavor of the apocalyptica so it's more cryptic and abandoned instead of dusty and broken, and don't worry so much about making it seem real. Add some surrealistic elements -- throw in horrible things that don't make sense, then watch the players creep themselves out trying to make sense of it. I learned this technique a long time ago when running Nobilis and it always works.
The Fronts are pretty good for cosmic horror already, just take their descriptions and go with weird and cosmic trappings instead of translating them into more mundane post-apocalyptic terms. If you read the names and impulses of the different threat types and take them literally, you'll get lots of nasty ideas.
The Maelstrom, if you're going with classic Lovecraftian horror, probably translates into the realization that the cosmos is indifferent and inimical to human life and sanity. You might come up with some nasty custom Moves to make up for the lack of a "SAN roll" in Apocalypse World, something to do with when you strive to understand the arcane logic behind the transportation system... or when you take anti-hallucination pills and see what's really out there...
Regarding threats, if you want to make them scarier, take some of Dungeon World's advice for making monsters scary. Google the story about the 16 HP dragon. The really bad stuff shouldn't be scary because it has a lot of hitpoints, it should be scary because it's awful and incomprehensible.
EDIT: Maybe look at Monsterhearts, too: beneath the teenage rebellion, there's a creepy game about people trying to control each other with unsuitable methods in order to meet their own needs. Plus the playbooks have a whole heap of weird powers that can be stolen and adapted.