The Amazon Appstore does offer free apps of the day, but today's is listed as Cogs, so The Cave's sudden free-dom is certainly a touch curious. We've not seen any official word from Double Fine on the matter, so we've reached out to the studio for clarification.
Double Fine puzzle-platformer hiked its way to three and a half stars in our review of the Xbox Live Arcade version: As Ludwig put it, "The Cave turns out to be the cruelest of Double Fine's games, and perhaps the most sharply written. Ron Gilbert and Chris Remo miss on the explicit narration, but their underlying story is perfectly pitched through puzzles and your own wicked participation."
Update: Free-dom is fleeting, it seems. The Cave is listed on Amazon now at the regular $5 price.