Temperature and pH gets more complicated than the effect on CO2 solubility - though CO2 is probably the only variable we'd need to change clinically.
Even in plain water, pH is 7.0 at 20 degrees C, but 6.8 at 37 degrees C, simply because more energy in the system = more dissociation of ions = more H+ (and OH-) = lower pH. We don't even have reference values for temperatures other than 37 degrees.
We can divide the effects of temperature into respiratory and metabolic components:
Respiratory
First, increasing temperature encourages gases to leave solution which
decreases solubility. So a cold patient can dissolve
more CO2 = more carbonic acid. This is catalysed by carbonic anhydrase so it is a quick process, and makes the patient have a lower pCO2 as the temperature drops (because the CO2 has been sequestered as carbonic acid).
Metabolic
So, where does acid-base actually matter? In the end, it's the intracellular machinery that cares most, and keeping the intracellular pH in a good place is the whole point of acid-base balance. The machinery that is actually affected by pH changes is the imidazole groups hanging off the end of the amino acid histidine.
The pK of these imidazole groups is
temperature independent. This means for a change in temperature we only really need to change pCO2, if anything at all.
Now, the last question to ask is - who cares? Some people evidently do, and they fall into one of two camps:
pH-stat
The 'by the numbers' approach. No matter what, pH must be 7.4, chemistry be damned. Admit to the ABG machine that your patient is 20 degrees C, and it will dutifully tell you that your pH 7.4 pCO2 40 patient is actually pH 7.65 pCO2 18. Panicked, you may add CO2 to your patient - which could actually improve CBF and outcomes in paediatrics...
alpha-stat
The intelligent approach. Imidazole groups don't care about temperature, and the normal pCO2 range changes with temperature. Luckily for us, when you re-warm the blood sample to 37 degrees in an ABG machine, the reference values at 37 degrees can be used. Just don't tell the ABG machine to un-correct these numbers by telling it your patient's actual temperature!
What does this mean clinically? I personally think it means just tell the ABG machine the patient is 37 degrees no matter the actual temperature (and thereby practice alpha-stat pH control). If you do complex paeds, you may find yourself telling the ABG machine the truth about how cold your patient is (pH-stat) so it can adjust the numbers for you and encourage you to add some CO2 to the bypass circuit.
Explained much better by the excellent Kerry Brandis:
1.6 Alphastat Hypothesis