Double Pain is something of a novelty badge that doubles the amount of damage that Mario takes. It can be purchased from Charlieton in Rogueport for 36 coins. It uses 0 BP.
There are very few strategically valid reasons to use this badge. The probable intended use of this badge is to partner it with the Return Postage badge creating a situation in which most enemies do significant damage to themselves as well as Mario.
A secondary use of this badge, in which is it temporarily used, is to use its function to quickly and intentionally reduce Mario's HP to below 5 (placing Mario in "danger") and then de-equip the Double Pain badge. Doing so would ensure that the various Close Call family of badges is made active reliably.
A third usage is an example of emergent gameplay that is entirely player preference. This function is to intentionally increase the difficulty of the game -- essentially functioning as a "Hard Mode" for players who are so inclined to take such a challenge. To use it this way, players seek out the Double Pain badge as quickly as possible, equip it immediately upon purchase, and never remove it.
It can also be used to break the rule "You can only damage 30 points" in Glitz Pit. This is necessary when a badly placed opponent has not yet been scanned.