Verse 8 tells us that Elijah was brought to Mount Sinai; this verse tells us that he stayed in the cave there. “The cave” (as opposed to “a cave”) suggests that it is one with which we are already familiar. The various commentators (Rashi, Radak, Metzudas David, etc.) all agree that it is the cave where Moses stood when G-d caused His “presence” to pass by, in Exodus 33:22. (See also Talmud Megillah 19b.)