The Green Millennium by Fritz Leiber

From a Grand Master of Science Fiction and Fantasy: Everything changes with the arrival of mysterious, green cat.

In a futuristic human society based on our own, the regimented, regulated, and bureaucratized lifestyle led by the misanthropic Phil Gish leaves him feeling vaguely dissatisfied and emotionally cut off from other people. When a green cat appears in his room, he is surprised to find that it makes him feel happier and more alive than he has ever felt. Phil decides to call the cat Lucky, hoping his life will take a turn for the better. But lucky is in high demand, and Phil quickly realizes that he's going to have to go up against the mob, the FBI, some nude aliens, and a gorgeous mystery woman if he's to keep him. When Lucky seems to vanish into thin air, Phil will do anything to get him back, even if it means challenging the very powers that rule his world.