According to the PKCS#11 spec, all key objects have defined a semantic for CKA_START_DATE and CKA_END_DATE, although, for key objects that are private (e.g. CKO_PRIVATE_KEY, CKO_SECRET_KEY), the retrieval of their values fail.
This is happening when an attribute with a specialized class, e.g. P11AttrStartDate, gets added to a private object, its value is always written in clear, due to the updateAttr method overload, although, upon retrieving the value, due to the retrieve method not being symmetrically overloaded, and because the object is private, the attribute value is decrypted and fails.