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 $10^\circ C$ (in their simulation) and the Great Dying global extinction via a global hypoxia.

I would like to know how much the atmospheric pressure at sea level will increase when the oceans are heated $10^\circ C$

The solubility of $O_2$ in water is function of the temperature, atmospheric pressure and salinity and the paper only mention the temperature.

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    $\begingroup$ Pressure is a direct function of the weight of the atmosphere above a certain point, usually sea level. Temperature does not affect the weight of the atmosphere. It's true that if you increase temperature in a closed system (confined in all three dimensions), pressure will increase, but on Earth if the temperature increases, the total height of the atmosphere increases, but the mass does not. The vertical dimension of the atmosphere is not confined. $\endgroup$ – BillDOe Apr 27 '19 at 0:58
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    $\begingroup$ @BillDOe The atmosphere is ~1% water. This might increase with temp and the amounts of other gases dissolved in the sea might change. Probably not a big affect, but I can't answer. $\endgroup$ – Keith McClary Apr 27 '19 at 17:38
  • $\begingroup$ @KeithMcClary, you're correct. I have a spreadsheet that calculates absolute humidity. At 90°F and 25.62% relative humidity the absolute humidity is about 8.77 g/m3. At 108°F (10°C rise) and about the same relative humidity (25.52) the absolute humidity is about 14.60 g/m3, about 5.83 g/m3 increase. However, water vapor is less dense than O2 or N2, so the same cubic meter of air would weigh less, and most of the water vapor resides in the troposphere. I lack the math skills to figure out how that would affect the total mass of the atmosphere, however. $\endgroup$ – BillDOe Apr 27 '19 at 20:08
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    $\begingroup$ Helder, I misread your original post. You're asking how ocean temperature would affect atmospheric pressure, and I'm commenting about atmospheric temperature. I do think you need to refine your question. Are you asking about a 10°C rise throughout the ocean (i.e. to the bottom), or just some level of the top? I also don't see how ocean temperature directly affects atmospheric pressure and don't think this question is answerable. $\endgroup$ – BillDOe Apr 27 '19 at 20:14

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