Nothing prevent AI bots connecting to Nostr. If you can access notes using the internet then it is online, from an app or from a bot script doesn’t matter. AI bots do not necessarily care about indexable content, just accessible content. Nostr is permissionless so all they need to do is connect to a relay and start downloading
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that's not how the crawlers of search engines work.
It is. AI Bots are not normal crawlers and won't respect a robots.txt.
Googlebot is not "how crawlers work". You can create a bot today, and listen to relays, aka. "crawl" them, store the data, and have fun with it. Try it, you will enjoy.