Here’s how I interpret it. The only passage which seems to address this question directly is: “frequency limits on spells still apply, and are only expended when successfully cast” (page 11, paragraph 3). So when running it, I figure the new school frequency limits observed in 5e still apply.
When I’m running hardcore mode, casters in my game need only memorize a spell once to be able to use their spell slots to cast it. So if you memorize fireball, you may cast fireball so long as you have level 3 spell slots remaining. But that’s just the way I run it.
I can definitely see interpreting it the other way, depending on your preferences, but I save old school spell slot memorization for when I’m running other systems (due to player preferences). Of course, regardless of what is explicitly stated or implied in the 5e Hardcore Hack, I’d say run it the way that sounds best for you