Hmm. Damn. I wanted to do something like this:
The first capital upgrade is the same as default. It forms the cornerstone for allegiance in the empire.
The second capital upgrade provides defensive bonuses. Extra health, the ability to recover under siege, passive culture generation, and a central PJI covering the system.
The third capital upgrade provides expansion bonuses. More health, extra logistics slots, some tac slots, and "Empire's Reach". This ability would colonize neutral planets under the influence of your culture every 5-15min. Local militia would probably make the ability pretty useless, so it'd wipe them out as well (but not pirates or players). This would enable a culture based expansion for the middle stages of the game. Advent can synergize the ability with their culture cannon late game, and it could find use for planets lost to the Novalith Cannon (though the capital is likely to go first).
The fourth capital upgrade provides empire bonuses. More health, a massive number of tac slots to fortify the world, extra fleet slots, extra cap ship slots, and an amazing 5% reduction in fleet upkeep. At 75% upkeep, the 5% reduction bonus would make up 20% of your income, from the capital alone(though it would be a much smaller relative impact at lower upkeep levels). The total upgrade path to here would be VERY expensive, and would be unlikely to pay off for all but the largest empires.