Anecdotal: for the specific case of "freshly written standup comedy routine", mental palaces [1] have proven effective, at least for me.
Once you remember that "your next joke is about x,y,z", it's pretty easy to remember the joke itself.
Once you remember that "your next joke is about x,y,z", it's pretty easy to remember the joke itself.
Like general public-speaking, approaches used by individuals to prepare/perform stand-up do seem to vary based on personal preference/comfort/style.
(In a similar way I wasn't a fan of/good at "rote memorization" in my academic life either.)