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Hello and thanks for using JustAnswer!
I am Derek, a master Volvo technician and i will be happy to help with your S80.
The most common cause for the problem you are describing is a failing fuel pump. When cold the pump will work normally and as it heats up it can cause the problem exactly as you described.
To test the fuel pump is very easy, there is a fuel pressure test port located on the fuel rail. You will need to rent or buy a fuel pressure gauge from a auto parts store.
I would recommend testing or replacing the fuel pump first.
If you have any other questions please let me know.