Pirates could be losing their base(s) to culture--if they lose all bases, they stop spawning raids.

If you play against AI (and if it's a 3-4 hour game, that sounds likely) typically the AI will novalith the pirate base(s) first. Also, they might 'decide' the pirate base is a valuable strategic position and throw their entire fleet at it. (I've seen similar behavior targeting non-colonizable gravity wells with resource asteroids.)

You might want to count your blessings anyway: if the colonization stage is over and the game is still going, most games get deadlocked with heavily defended chokepoints, so a pirate raid can actually be quite helpful to the target. Pirate ships--which have no support abilities or strikecraft--are easy kills if they hit a fortified world, giving the target cheap cap ship or titan experience. (AI will also have salvage researched for easy resources, and some factions can steal ships.)