Google's John Mueller wrote on Google+ that both browsers and crawlers support HTTP/2 web server configuration and he recommends that if you have a choice, you go with it. But he made it clear that going HTTP/2 with your server does not have a direct SEO benefit.
HTTP/2, as John explained, "doesn't change the URLs, and is transparently supported for users & crawlers if they ask for it." It does help improve the performance of your host, so it does speed things up. He said "you don't have to set up redirects, change links, add markup, or make any changes in Search Console."
He did clarify that GoogleBot won't yet support crawling HTTP/2 specifically yet, which is surprising because we thought it would be here by now. But even if you are HTTP/2 he said search engine crawlers, including GoogleBot "will continue to work normally."
John then added on Twitter, "HTTP/2 is for speed. Fast sites make users happy, happy users recommend websites, so it's very indirect."
Have any of you gone HTTP/2?
Forum discussion at Google+.
