The question is essentially does the Sun Blade double smite damage versus undead?
Relevant quotations:
Against Negative Energy Plane creatures or undead creatures, the sword deals double damage (and ×3 on a critical hit instead of the usual ×2).
If the target of smite evil is an outsider with the evil subtype, an evil-aligned dragon, or an undead creature, the bonus to damage on the first successful attack increases to 2 points of damage per level the paladin possesses.
Assuming a normal damage output of 1d10+10 on a level 20 paladin character, is the first hit vs. undead doing 2d10+60 and 2d10+40 on subsequent hits?
The Sun Blade text says the sword deals the increased damage. This seems to be inclusive language, meaning that is relevant to the sword (and only the sword), more specifically the weapon damage dice listed, and is supported by the factor the critical multiplier of the sword itself increases (and not just the wielder's multiplier). If it said the wielder deals double damage, you would be correct about Smite Damage and the such being multiplied.
That being said, if there was Vital Strike involved, the enhanced damage dice from the Sun Blade would apply to the multiplying of that (to 4D10, 6D10, etc.)
The Sun Blade text says the sword deals the increased damage. This seems to be inclusive language, meaning that is relevant to the sword (and only the sword), more specifically the weapon damage dice listed, and is supported by the factor the critical multiplier of the sword itself increases (and not just the wielder's multiplier). If it said the wielder deals double damage, you would be correct about Smite Damage and the such being multiplied.
That being said, if there was Vital Strike involved, the enhanced damage dice from the Sun Blade would apply to the multiplying of that (to 4D10, 6D10, etc.)
Mind you, smite is just a bonus to the damage of your existing attacks. If it were a distinct source of damage that just happened when you hit, I would agree with your reasoning. However, it's not. I'm not, though, asking if it universally doubles smite damage vs. undead if you were to use a different weapon. A level 20 paladin smite punching with their fist would do just +20 damage even if they also had a Sun Blade in the other hand.
Darksol, what do you think happens when you charge with a lance? What gets doubled? Only the dice? Because...
So, yea... A weapon dealing double damage seems to mean attacks with that weapon do double damage - at least that's always how I've seen it played. That's why things like vital strike call out doing twice as many dice, as opposed to "double damage."
Benefit: A lance deals double damage when used from the back of a charging mount.
Honestly, that just tells me the RAW regarding the Lance is grammatically incorrect, especially since it fundamentally follows the same wording as the Sun Blade. If the intent is that it doubles all damage regarding a given attack, the RAW would need to be worded to something like this:
Benefit: Attacks made with a lance while on a charging mount deal double damage.
The Sun Blade would need similar revisions, like so:
Attacks made against undead or creatures from a Negative Energy Plane deal double damage (and have a x3 critical multiplier instead of the usual x2).
With that being said, I understand the intent behind it, and since a Lance, a mundane weapon, has the intent of doubling all damage associated with charging, it's not a stretch to assume that same intent is followed with a high-end magical sword. But again, the RAW does not reflect that sort of intent.