When my opponent has Goodra with the "Slimy Room" Ability in the Active Spot, if I try to attach an Aurora Energy to one of my Pokemon and I flip tails, do I have to discard a card for the cost of Aurora Energy too?
No, since the flip happens before it is attached, if Aurora Energy is discarded you do not have to discard another card along with it.
Source: Fusion Strike FAQ; TPCi Rules Team (2021-11-11)

Relevant cards

Note: All applicable cards may not be listed below.
SWSH8_197
SWSH8_197
SWSH1_186
SWSH1_186