Rod of Absorption

Each Rod of Absorption tracks the cards it has exiled separately. If Rod of Absorption is destroyed and then later brought back to the battlefield, it will not be able to access any of the cards that it had exiled before.

If you cast a spell “without paying its mana cost,” you can't choose to cast it for any alternative costs. You can, however, pay additional costs. If the card has any mandatory additional costs, you must pay those to cast the spell.