I think it would be sense that the browser should never send cookies if the certificate changed unless the user specifies that the new certificate is valid for

I think it would be sense that the browser should never send cookies if the certificate changed unless the user specifies that the new certificate is valid for the existing cookies (or manually copies them if necessary, e.g. due to a different domain name or a different computer). This way, it willl be secure. HSTS is no good, but cookies being secure-only can help (although I dislike that cookie flag and think instead, it should be if the connection that sent the cookies to you is secure instead). I also think that, to save energy (and to work-around compatibility issues), you should be allowed to configure it to use insecure proxies for secure connections (usually these insecure proxies would run on your computer, or LAN, but this is not strictly required).