A certificate has a specific expiary date and if it got untrusted by the CA in this time (e.g. its private key has been leaked) you want to know that.
CRL (Certificate Revocation List)
Contra: Downloading the CRL generates traffic.
OCSP (Online Certificate Status Protocol)
Contra: The OSCP Server knows your requests.
OCSP Stapling (Online Certificate Status Protocol Stapling)
Contra: Nothing yet.