Check your antivirus software
Antivirus software interruption is one of the most common causes of the Apex Legends not working problem not starting. To check if this is your problem, try including Apex Legends in your antivirus or just turn off the entire program temporarily.
If this works for you, you should be able to start your game successfully.
Run your game through the EAC launcher
Starting your game through the EAC launcher is another way to start your game. To do so, simply go to your game’s installation directory, then right-click on Easy Anti Cheat_Launcher and select Run as administrator.