When playing double time, it shows the second automatically played card to the enemy player to reveal what secret has been played. I think.
the interaction with secrets already in played has always been like that. when a random effect tried to cast a secret alredy in play it will attempt and fail because you can only have one of each secret in play, the failed activation will be visible to the opponent. so when you play a secret with double time it will play the secret, and then play it again. the second attempt will fail and be visible to your opponent
This seems like a bug to me, ruining your strategic advantage when you don’t have any kind of choice
double time only cast your first spell twice, cast the secret as your 2nd spell if you want to keep it secret 