Why does an AS cube rebuild use so much TempDB?
While the cube builds, you can run Adam Machanic's sp_WhoIsActive diagnostic tool to see which queries are allocating space in TempDB.  I recorded an sp_WhoIsActive tutorial video to show how it works.  Include the @get_plans = 1 parameter when you call it, and you'll also get the execution plans.  That way you can see exactly what's using TempDB and why.