Cards that are in cooldown can't be listed for rent or buy until after cooldown is over.
I thought this was the rule, I know I can't rent out my cards if they are on cooldown. But maybe there's a way and I never found it :)
@unitqm I'm not the best person to vote on this, I don't know how those types of details work. But I do support helping make the game better to understand and educating players for sure.