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Changes in climate sensitivity as a function of Temperature

So apparently radiative forcing going like log CO2 has a name … https://en.wikipedia.org/wiki/Svante_Arrhenius#Greenhouse_effect though the explanation is somewhat complicated… See

Also, from Wikipedia, we have

\Delta T = \lambda \Delta F

where the climate sensitivity \lambda \approx .8 K/(W/m^2)

and we can approximate the logarithmic response of radiative forcing to changes in CO2

\Delta F = 5.35 \log \left ( \frac{P}{P_0} \right)

Combining these we get \Delta T = 4.3 K \log \left ( \frac{P}{P_0} \right)

T = T_0 + 4.3 K \log \left (\frac{P}{P_0} \right)

P = P_0 e^\frac{T-T_0}{4.3 K}

and \frac{dT}{dP} = \frac{4.3 K}{P} = \frac{4.3 K}{P_0} e^{\frac{-\left(T-T_0\right)}{4.3 K}}

So the climate sensitivity is not constant - but falls off by a factor of e every 4.3K change in temperature.