You probably can, but you need to be upfront about what you're collecting and the context that is being stored with it. It MAY be more ethically permissible to

You probably can, but you need to be upfront about what you're collecting and the context that is being stored with it. It MAY be more ethically permissible to degrade the context and preserve only the most valuable and least personally identifying data. (Such as saving only the actual search query and a local timestamp, but filtering out anything related to a recognized name that isn't famous)