Tempus is a ghost from an alternate future where he has taken over the world.
According to the alternate future, Tempus was infamous for leading the Great Spirit Uprising after causing an incident in the Containment Unit in 1997, thus leading to the Great Ghost War. He was able to take over the Earth by leading his ghost army and enslaving all of humanity, leaving a band of a freedom fighter to fight against his reign.
The last remaining band of freedom fighters rigged a Ghost Bomb and Proton Cannon from what notes, journals, and other records they could salvage to defeat Tempus. One freedom fighter named Isaac thought he destroyed Tempus with the Ghost Bomb, though this wasn't the case when it actually causes a Time Slip, which separates Tempus into two forms. Kylie Griffin and Isaac swapped places in the time stream.
The Ghostbusters attempted to trap the past Tempus, but he escaped the Ghost Trap and fled the Firehouse in confusion. He wandered to Times Square, where his future self-communicated his mission to complete the uprising. In the future, a prophecy foretold the return of Kylie Griffin and she would deliver humanity from Tempus' reign of terror. The Ghostbusters attempted to rewrite the future and send a Proton Pack to Kylie.
Both Tempus and his future self-tried to prevent this, but Kylie was able to secure one thanks to Garrett's quick thinking. She followed instructions left in her diary to confine Tempus as the site of the Firehouse. Once the others did the same with the past Tempus, the Time Slip reopened and Kylie merged the two in the present. The Ghostbusters then successfully trapped Tempus and the Time Slip closed, restoring all to their proper place, and averting the bad future for good.