No, hived only checks if json field of custom_json is a valid UTF-8 JSON string. It does not build variant object out of that string in order to access its fields. To see if it might need to actually do that, it uses id field that is right there in the operation itself.

Moreover the execution time associated with it being custom_json is trivial in comparison with cost associated with the transaction. Validation, execution and RC handling of custom_json (yes, cost of calculating RC is also included in the cost of operations) all together take 1509 ns on average. On the other hand duplicate check, tapos check and charging for RC takes 6622 ns plus a whooping 94165 ns per signature. Transaction itself also costs some state (128 byte-hours) and history bytes (size of transaction). As you can see, even if we started to charge zero for execution of custom_json_operation, the difference wouldn't matter.

If you are thinking about looking for a way to speed up calculation of public keys from signatures (most obvious target when it comes to execution time), it would help all transactions and would be very welcome, however most of the cost of transactions with custom_jsons comes from history bytes, not execution time.

_{The timings are obviously constants - they don't depend on the hardware the node is run on, because otherwise nodes would constantly disagree on RC mana of users. They were collected by @gtg on average computer recommended for witnesses - see issue #205. They are likely to be updated (lowered) with future hardforks as more powerful hardware is required to run witness node.}