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)

or

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

or

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.