However heat transfer from heatsink to air is not limited by the heat capacity of the air. It's limited by the various convection relationships and by the temperature gradient. The convection relationships are very much affected by airflow past the heatsinks.
This is before you consider how much above atmospheric pressure the air in a computer case can be. While air may move around it swiftly, the difference in density between various reasons is going to be slight, and heat capacity related to the absolute pressure not the gauge pressure.
So, I think you have a negligible pressure increase, which would be exerting a negligible effect on heat transfer even were the pressure difference considerable, and which is completely disguised by the convection relationships.
Unless you don't actually mean pressure, but as an engineering student I'd hope you know what pressure is.
