There's different senses of the word information, so maybe a cosmologist would hesitate answering this until you explained exactly what you meant. In objective collapse models, it is just not conserved because collapses of the wave function don't conserve information. I suspect though that what Neil has in mind is not micro physical information that gives us the exact micro state of the universe but something more like accessible macroscopic information. Such information can be lost and gained depending on different physical processes.
Welcome to the July 2023 Ask Me Anything episode of Mindscape! These monthly excursions are funded by Patreon supporters (who are also the ones asking the questions). We take questions asked by Patreons, whittle them down to a more manageable number -- based primarily on whether I have anything interesting to say about them, not whether the questions themselves are good -- and sometimes group them together if they are about a similar topic.
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Blog post with questions and transcript: https://www.preposterousuniverse.com/podcast/2023/07/03/ama-july-2023/
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