I wouldn't change the fuel pump just yet, just asking you to prime the system up a little more.
Near the fuel filter is a primer - you squeeze it and it brings fuel up from the tank. This is useful when you run out of fuel or change the fuel filter.
Normal operation is 3-4 squeezes and the pump becomes firm.
Small leak in system might need 5-6 squeezes before the pump is firm.
Major leak in system and it could take 20 or more.
It is also useful to find a leak that allows air into the system. Prime it more than you should, the pressure in the hose rises and fuel starts coming out of the leak. Fix the leak and the cold start problem might go away.
It might be something else, but we start with things that don't cost a lot to do.