John Mueller of Google said the other day in his hangout at the 1 hour and 13 minute mark that Googlebot still is not crawling in HTTP/2. The reason it is not is because Googlebot is pretty good at crawling quickly and it wouldn't take advantage of the same benefits a browser would benefit from with HTTP/2 he said.
Here is what John said:
No, at the moment we don’t crawl HTTP/2.We are looking into what we can do with that. In general, the tricky part there is that for the most part Googlebot isn’t the same as a browser so you wouldn’t see the same speed effects as a browser would in regards to HTTP/2. So we can cache things a little bit differently, we can do requests in a more parallax way. A little bit different than an average browser would do. So we don’t see the full advantages of Google going to HTTP/2.
But espesially as we see more web sites implementing the push functionality, HTTP/2 where you can request the HTML page and it includes all of the embedded content right away, then that might be something where Googlebot engineers say now it really make sense to actually implement HTTP/2 for Googlebot.
That does not mean you can't make your site HTTP/2, Google said go ahead and do that. We were however expecting Googlebot to support it sooner but I guess it is back burnered now. HTTP/2 doesn't give your site an SEO advantage nor a super big crawl advantage yet.
Here is the video embed:
Forum discussion at YouTube.
This post was scheduled to go live today but was written earlier - I am currently offline today.