Having the browser be able to track and tell me that "Though we aren't sure this is actually google.com, we do know that the exact same cert has been used the last 50 times you visited this website" is something I'd consider to be useful. (Actually, telling me if it changes would be the useful bit).
That would be at least be useful for self-signed certs (though those aren't really needed in light of Let's Encrypt...)
I'm curious. Has anyone ever encountered that scary warning you get when an SSH host key changes, and thought "oh man, I'm getting MITMed, I'd better not connect to this server!", instead of thinking "oh right, I guess they reconfigured the server, now what command do I type to make the warning go away"?
On the server side it's better for each server to have it's own private key and certificate which is valid for a short period of time and frequently renewed. So the compromise of one server does not compromise certificates on any other servers and the useful lifetime of a compromised key is very limited.
I think DNSSEC and DANE is the best solution. Allow the certificate thumbprints to be published securely in DNS. At least then we reduce the number of trusted authorities to the TLDs and the scope of authority for each one is automatically restricted to it's own TLD.
> Having the browser be able to track and tell me that "Though we aren't sure this is actually google.com, we do know that the exact same cert has been used the last 50 times you visited this website" is something I'd consider to be useful. (Actually, telling me if it changes would be the useful bit).
Isn't that what you do when you make a security exception for a self-signed certificate? Having that enabled by default lulls people into a false sense of security.