You have to find 4 special artifact pieces and bring them to the special room in the crown vault. They're scattered around in different areas in the game and the artifacts are saved between runs if you manage to place one where it belongs. I'll spoil them below, so it's your choice whether or not you want to know the exact answers.
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-Spoilers-
1st Piece - Enter the crown vault in an area where the low HP music is playing (Desert, Scrapyard, Snow City), the Artifact will most likely be there
2nd Piece - Go behind the throne in the Throne Room, there's another artifact right behind it
3rd Piece - Enter the warp zone and find the hidden area with the Twins pet. The artifact will be just behind them
4th Piece - Go to the IDPD base and check the side walls, after standing next to them for a bit, they'll open up and one of them will have IDPD soldiers and the other one will have the last artifact. Then you have to defeat the Captain and use the new pet that she drops to warp back to your original run
Bring each of those artifacts back to the crown room and prepare for the worst.
Good luck!