Without getting into much detail, gases have a saturation vapor pressure that varies with temperature. To have a saturation vapor pressure there has to be a liquid or solid present with the gas. If the concentration of the gas cannot vary, i.e. a balloon full of gas that is warm enough not to condense will continue to have just as may gas molecules as liquid/solid molecules if they don't react with the material that the balloon is made out of. If you happen to have a butane lighter, you know the flame stays about the same as long as there is liquid butane sloshing around inside the lighter. There is some variation in the flame based on the temperature of the lighter, but as long as there is liquid that can change to gas, it lights.
CO2 is a gas that can be absorbed in water. Depending on the temperature of the water and the air, the amount of CO2 in each will vary. There is a lot more CO2 in the water than the air, so the temperature of the water is a stronger control variable.
For water at 0 C degrees, the saturation vapor pressure of CO2 is about 3.4 g (CO2 gas) per kilogram of H2O (liquid). At 4C degrees, the saturation vapor pressure 3.0 g (CO2 gas) per kilogram of H2O (liquid). A 4 C change in the water temperature would produce a 10 to 13 percent change in the amount of CO2 the water could hold.
Since I tend to run on about the changes made with the Drake Passage opening, I am not going to lose another opportunity to run on about the Drake Passage opening. That opening cooled the southern hemisphere at the expense of northern hemisphere warming with an approximate net decrease in global average temperature of 3 C degrees. That happened because the Antarctic Circumpolar Current effectively isolated the Antarctic thermally from the rest of the world. Because of the more efficient mixing of the oceans allowed by the 100 to 150 million cubic meters or water per second continuously circulating that wasn't circulating before, some things changed. One was the "Average" temperature of the Southern Oceans which make up the majority of the global oceans. Since colder water can hold more CO2 and colder water sinks to some deeper depth in the oceans, the relationship between atmospheric CO2 and ocean CO2 changed. It is really pretty simple physics.
Since the "Average" temperature of the colder surface water in the NH is about 3C warmer than the colder surface water in the SH, there are different rates of CO2 uptake and out flow from the oceans. About 10 to 13 percent difference in fact, if the oceans and atmosphere are in some true equilibrium. If the oceans and atmosphere are not in an equilibrium state, then the temperature changes the rate of uptake or out flow of CO2. There is other stuff going on, pressure is a factor, so the deeper the cold water sinks with its CO2 captured at the surface, the longer it can hold onto that CO2. Then since CO2 forms a weak acid in water it can react with the bases in the water either strongly, like rock or shells or weakly like various carbonate, or carbonites or whatever happens at different times in the Marine Carbonate Chemistry cycles. You can make a career out of that stuff all by itself.
If the average temperature alone can make a 10 to 13 percent change in the CO2 "balance", it would seem likely that the other cycles would respond when concentration near their "sweet spot". Which when you consider that the minimum temperature of the sinking cold sea water is closer to -2C degrees, if you have enough time on your hands you could "explain" a natural variation of CO2 on the order of +/- 50 percent.
There are a lot of smart folks working on these potential impacts and I am not one of them. I just want to bring the temperature portion of the puzzle to the fore front for a moment. If the average temperature of the oceans were to change 4C degrees, the total change in carbon dioxide would be on the order of 1 x 10^20 grams. That is one serious butt load of carbon dioxide. If it could change that much, it probably would have already. That implies that the Carbon Cycle is much more interesting than just temperature relationships.
Mankind is one of the biological components of the Carbon Cycle. We do have some impact, we just are not really smart enough to know how much, at least yet.