As Hüsler and Sornette discuss here, the world's CO2 emissions were superexponential, up until around the middle of the 20th century when the population growth rate peaked. Before that, a realistic-looking model for the amount of CO2 in the atmosphere over the last millennia is
where t is time, going from 0 to 1, and C(t) is the atmospheric CO2 concentration at time t. (More on the exponent 2/3rds below.) This means C=1 at the start time t=0, which you can rescale to 280 ppm if you want, but I'll keep the function simple to make the conclusion clearer.
Then the forcing from that CO2 in the atmosphere is
and the change in temperature, which is proportional to the change in forcing, is
(I've dropped proportionality constants.) This is simple enough. I've plotted these functions to the right.
Again, I'm not interest in an exact match, only getting a realistic shape for the C(t) curve. (You can rescale everything by adding constants and multiplying constants, if you really want to get a good match to the observed values.)
The curve C(t) is superexponential -- it's increasing faster than an exponential function. The green line is the exponential fit to the C(t) data points using linear regression, and it can't keep up with the C(t) curve.
The brown line is the resulting temperature change -- a hockey stick. It's a straightforward consequence of the world's path on CO2 emissions and the basic physics.
Thus, it would be surprising if any of the paleoclimate studies gave anything other than a hockey stick.
Of course, the real world is messy with natural flucuations and nonlinearities and the like, so a hockey stick isn't guaranteed by the data. But it seems to me a hockey stick is the best, first guess. (I think it was John Wheeler who said you should never start a calculation until you know the answer, and this, plus a good intuition for the actual numbers, is the kind of thing he had in mind. Though few of us are John Wheelers, and this kind of argument shows up for me usually only in retrospect.)
In their paper, Hüsler and Sornette construct a very simple economic model that gives this superexponential for atmospheric CO2. They make some basic and plausible assumptions -- about population growth, the labor force, about the amount of capital, and about the progress of technology, and get a set of coupled differential equations they solve. A simple case give the exponent 2/3rds I used above. You can tweak that too, if you want. See their paper for details.
C(t) is no longer increasing superexponentially. It can't do that forever, in a real, finite world -- as Hüsler and Sornette write, such an increase eventually leads to "regime change" -- a fundamental change in an input function like population. Perhaps that's what the peak population growth rate in the 1960s was.
On the other hand, climate feedbacks are coming into play, so the temperature increase is hardly over yet. Really, it's just getting started.