The endgame of a wisely fought War is supposed to be Peace.

Sun Tzu vs Genghis Khan, is this correct or is the aim to accumulate more and more to fight more and more?

You don't mean 'endgame'.

Genghis Khan's quote was a respomse to a letter the catholic church sent him that was filled with religious mumbo jumbo. You should read the churches full letter and his full response after learninf the context. 

I can tell you Sun Tzu's advice is good for most societies. It is usually better to avoid war or to win through other means. Ghengis Kahn had bo such luxury. The obly way he could keep his people unified was through constant war as was evidenced by his predecessors who lost their alliances of the mongols when their reasons for unifying were taken away. It was also just a fact at that time, societies that refrained from war had poorly skilled fighters and were more easily conquered.

Sun Tzu's advice to a unified single people was corrext Ghengis Kahn's tactic of constant war was also correct but he was trying to transition before his death to a more stable society by implementing things like public education. Unfortunately his kingdom was ripped apart by internal warring factions between his children who were warned what would happen if they were not unified by their father

Yep.

Endgame is definitely the wrong word.

Aftermath might be more appropriate.

The objective of warfare is always the same. To impose ones will upon the enemy.

The 'ideal aftermath' for making said imposition depends on the nature of the victor's will.

This particular quote was allegedly spoken to Muslims after the conquest of Bokhara, so it was not actually directed at Christians.

That said, the Mongols did certainly have interactions with Europe and even The Pope himself, but this is particular quote is not from this interaction.

Oh, well I confuse it with a response he made to catholics which I found amusing. I'll look into this i teraction you mention more.