Just a heads up, this process began in a limited way in September and November of 2020. Now, according to Gary Illyes, “the h2 crawler traffic is gradually ramping up.” He said you may get a notification over the next “few days” as he just initiated that notification in Search Console.
Here is the screen shot of the notification he shared. It reads “Google will start crawling domain over HTTP/2.” It goes on to say “Google systems have determined that domain supports HTTP/2, so most Google crawls for your site will be over HTTP/2. You may see this change in your site’s logs. Google has decided to use HTTP/2 when crawling sites that support it, because HTTP/2 can improve crawling efficiency, both for Google and for your site. This change has no impact on your site’s rankings, and no action is required from your side.”
Here is the screen shot:
A number of site owners are now receiving this message from Google.
Again, this does not impact rankings but can save you hosting cost if you have a massive site. Gary reiterated this:
the h2 crawler traffic is gradually ramping up, it's not like you got the message and suddenly all crawl is h2. it may take a few days
— Gary 鯨理／경리 Illyes (@methode) January 13, 2021