Actually this is a use-case supported by this RFC [1]. You accept an arbitrary QUERY /search/ and you cache it on your side (or in a middle box somewhere such as a CDN edge) you can return in your response:
Location: /search/?queryHash=SOMECDNHASH
Sure, it is more work for your webserver to compute that and potentially the browser to cache it's knowledge of that QUERY, but it potentially gives you an advantage in keeping things like CDN edge caches generally aware of client/browser caches in a way that can be performance optimized.