I bought a managed IAP (e.g. the Coin Doubler) from Google Play, but how do I get that again on my new device?

If you are logged into the Play Store with the same account you used when you originally purchased the item, you should be able to click the item (e.g. Coin Doubler) again, and it will give it to you without re-charging you.

Please note that only managed items like the Coin Doubler or Go Premium can be restored.  Consumables like coin or gem packs cannot be restored.