Does difficulty adjustment apply when you mine in secret? Imagine you have an enormous amount of hashpower (say a nation state level attack) and you mine in secret. Does the difficulty adjustment apply specifically to your secret chain?  Because that would mean that you need to add hashpower at a faster rate than the public chain is adding it for your attack to effective?
Yes. The difficulty adjustment is part of the consensus rules. In order to create a valid chain that you intend to use to replace the current main chain, you must follow all of the consensus rules. That means adjusting the difficulty in the same way any verifier would be when they validate your chain.