Climate sensitivity (S) is the change in global mean surface temperature for a doubling of atmospheric CO2 level.
If you want to do a simple calculation, you have to make some assumptions. Mine are:
1) "temperature" and "CO2 level" are the 10-year moving averages.
2) temperature change is proportional to the logarithm of the CO2 level. So
ΔT = T - T0 = k ln(C/C0)
where k is some constant, C denotes the atmospheric CO2 level, and the subscript "0" denotes the starting point (or really, any reference point). If you plug in C=2C0 you find
k = S/ln(2)
so you can calculate the climate sensitivity S between two points in time t1 and t2 by
S = ΔT [ln(2)/ln(C2/C1)]
where ΔT = T2 - T1. (This result isn't anything new, of course -- I've just presented it for full background.)
Again, I'll assume the "temperature" at any time is the average of the last 120 months up to that time, and the same for "CO2 level." Then for the Hadley Centre's HadCRUT3 surface dataset and for the UAH lower troposphere (LT) dataset, and using the seasonally corrected monthly CO2 levels from NOAA, I find
S(HadCRUT3) = 2.0°C
S(UAH LT) = 2.1°C
which agree better than I expected.
(Yes, I know the LT isn't the surface, but I thought I'd try it anyway.)
The first is calculated over the 44-year time period of Feb 1968 to today, because the CO2 dataset starts in March 1958. The second is over the 23-year time period November 1988 to today, because the UAH dataset starts in December 1978. If you instead use the 5-year moving averages, I find the current climate sensitivites are 1.5°C and 1.4°C, respectively, but they fluctuate by several tenths of a degree depending on what end period you pick. Other choices of temperatures, CO2 levels, and datasets are left as an exercise to the reader.
Of course, there are all kinds of IABs here (if, ands, and buts). I've ignored all other factors that influence climate, particularly aerosols. They're in the data, and may be keeping the planet cooler than it would otherwise be, but keeping the air polluted (and having to add more) hardly seems like a solution to the global warming problem (unless it involves, say, geoengineering by nonharmful aerosols put there on purpose, such as marine cloud brightening).
And, of course, feedbacks may just be starting to kick in, like ocean saturation, the ice-albedo effect, Arctic methane emissions, etc.
And temperature change may be more than (and more complicated than) log(C) for higher values of C. See, for example, Appendix B of the famous 1988 Hansen paper.
The IPCC finds S = 2.0 - 4.5°C, and the recent Schmittner paper in Science finds S = 2.3°C with a range of several tenths of a degree.
Is a climate sensitivity of 2.0°C worrisome? That, too, is left as an exercise for the reader. Right now CO2 levels are increasing at about 0.5% per year, up from about 0.4%/yr in 1975. That would imply CO2 levels will be about double the pre-industrial level of 280 ppm by about the year 2070.
US readers: remember, a change of 2.0°C is 3.6°F.