can you install an exhaust system on a car when it has just been turned off?
By the time you get the car on a lift, the pipes have cooled off a lot. The only part that stays hot for a while is the Catalytic converter. The exhaust manifold will hold heat for a few minutes too. If you are doing exhaust pipes, you will be using a torch, so you will be working with hot parts anyway. Be prepared.
Since you would probably have to touch or remove the current exhaust system to install the new exhaust system, it would be better to wait until the car had time to cool down. The exhaust system piping gets very warm and could burn you. If you do try to change it right away, be sure to wear heat resistant gloves and long sleeves to protect yourself from burns. (the pipes would feel a lot like touching a hot oven.)
If you want to burn your hands