How much increase in the atmospheric pressure when the global temperature is elevated by 12 C?

The deep-sea temperature was $$12º$$C higher $$50$$ My ago.

I would like to know how much the atmospheric pressure at sea level had decreased and how much the water-vapour changed.
One can consider no ice-sheets and that the sea surface temperature varyed by the same amount and that by now there is $$0$$% of water vapor in the atmosphere (in fact it is between 0% and 4% function of latitude and altitude and assume the conservation of atmospheric masses of $$O_2, N_2$$).

This issue is very important because in this paper:
"Temperature-dependent hypoxia explains biogeography and severity of end-Permian marine mass extinction" is established a link between an increase of the global temperature by 10ºC and the Great Dying global extinction via a global hypoxia. The solubility of O2 in water is function of the temperature, atmospheric pressure and salinity and the paper only mentions the temperature.

This is a call to moderators to return this question to the PSE site where it was originally formulated.

This question is specifically formulated in the field of thermodynamics with matter subject to a gravitational field and the most appropriate site to get a correct answer is the PSE as the numbers clearly show: On the Physics website (PSE) there are 8,489 active issues marked with Thermodynamics and in this EarthScience site there are only 122 questions.   I will offer a bounty of 500 points as a reward for the first correct answer after it is returned to the correct site.

• Why do you feel that the total atmospheric pressure matters? It is only the oxygen partial pressure that determines its solubility.
– Chet Miller
Apr 29, 2019 at 11:11
• This is tricky because the atmosphere isn't quite like a gas in a box. Its "container" is just the Earth's gravitational field, and the "size" of this container (i.e. the atmosphere's scale height) can change with the kinetic energy of the gas molecules. If you heat up some gas in a fixed box, then the pressure increases. If you heat up the atmosphere, then the average height of the atmosphere increases, with effects on the pressure that are not intuitive.
– probably_someone
Apr 29, 2019 at 11:22
• This would probably be a better fit on Earth Science.
– JMac
Apr 29, 2019 at 11:45
• @BillN if temperature was increased now, water molecules would be added to the already present molecules in the atmosphere, they would not be replacing heavier molecules. Additional molecules may cause expansion of the atmosphere, but still will make the atmosphere heavier.
– Ján Lalinský
May 7, 2019 at 11:44
• @BillN yes but net mass of atmosphere would increase due to addition of mass. Pressure is strongly correlated with mass of atmosphere, not its density.
– Ján Lalinský
May 8, 2019 at 21:21