Early on a large part of your score comes from population, so a high score could be an indication that you are not whipping enough. The AI doesn't whip enough and grow their cities large. On immortal or deity you should rarely be at the top of the scoreboard in the early game, unless you conquer your way to twice as many cities as anyone else has.
There's a -1 diplo modifier you can get from smaller civs saying they are afraid of your large empire, or something like that. If you're at the bottom half of the scoreboard, the other civs at the bottom should give you a +1. But these are still very minor modifiers. There are much larger modifiers to be gained from trading and picking the right religion. But you should always keep an eye on what they think about each other as well. Trading with the wrong AI can give you a -4 "traded with our worst enemy" modifier with multiple civs. Usually you need to pick sides and can't befriend everyone.
Your power rating only acts as a yes/no switch for the AI when they consider war. Either you are too powerful, in which case they don't plan a war against you, or you're not, in which case they can decide to start plotting. If you aren't, it doesn't matter if you are 1% below the threshold or only have one warrior in your entire empire. On the higher difficulties it requires significant investments to reach a high enough power rating, so it's better if you can control it through diplomacy. Or maybe divert their attention to someone else by bribing them into a war.