DAVdroid uses okhttp as an HTTP client, and okhttp is ready for HTTP/2. We’re trying to contribute to okhttp by submitting qualified bug reports, patches, components (Digest authentication handler) etc.
Major HTTP/2 (RFC 7540) improvements (in comparison to HTTP/1.1):multiplexing of requests over a single TCP connection, thus saving costly TCP handshakes (= time and traffic), saving costly TLS handshakes (= time and traffic); more efficient message processing through binary message framing, compression of HTTP headers.
In CalDAV/CardDAV context, it means that for synchronization of one collection, only one TCP/TLS handshake is required and HTTP headers are transferred compressed. (Body compression using gzip is also supported.)
Many servers are supporting HTTP/2, for instance nginx 1.9.5+ and Apache 2.4.12+.
Our Web sites, including this forum, support HTTP/2, too.