Sometimes, CAs get compromised, like Diginotar or Comodo. In such an incident, anything can happen from individual certs being revoked to the entire tree being scrubbed from the cert store. A company that isn't aware of this in time might find themselves without a trusted CA.
In case this happens, does TLS allow for a website to basically have a backup certificate in case the other one is revoked? Or does that lead to other security issues?
To be clear: I'm not talking about having multiple root CAs validate your website with the same cert to have more trust. I'm talking about having a backup certificate, already deployed to the website, that's automatically checked in case your default CA gets revoked, so your website always has trust.