I think you just volunteered yourself! Gem cost goes up a gem every 2 hours. ie - 2 hour cooldown is 2 gems, 4 hour is 3 gems..
The cost per level of upgrading is the same on helm, chestguard, cloak, and relic. Spirit is different, and not sure on weapon.
Now, go start tracking the costs and cooldowns and let us know when you have that spreadsheet.