The basic premise isn't wholly unreasonable. Their overall goal is to diagnose sensitivity from the paleoclimate record. Essentially, their paper hinges on the premise that they have two major unknowns to be estimated - climate sensitivity is one, and the other is the forcing effect of a given atmospheric dust loading (aerosol). They simultaneously solve for both of these factors by using two intervals in the paleoclimate record over which the changes in forcings, and the temperature response, are reasonably well known. The intervals they use are (1) the strong warming from the Last Glacial Maximum (21 ka before present) to the Holocene, and (2) the rather smaller temperature change between the LGM and a warmer period which is observed about 41ka BP. There are observations of the changes in major GHG concentrations across these two intervals, also changes in dust concentration, and observations of temperature, all from the Vostok ice core. Their analysis generates a very large forcing estimate for dust (about 3 times as large as the standard estimates), and simultaneously a small estimate for climate sensitivity of 1.3-2.3C (95% probability interval).
The most glaring problem is probably in the bizarre and strongly biased interpretation of the temperature data. I've reprinted their figure here, with some colour annotations of my own:
The main feature to focus on is the solid wiggly line showing temperature from the Vostok core. The authors claim a temperature difference of 10.2C for LGM to Holocene, and 4.8C for LGM to 41ka BP. They do this by looking for the very lowest single data point in the LGM epoch, and comparing this to the highest peaks in the other eras. There is no attempt at any justification of this choice. I don't know exactly which data points they used, but based on the numbers they quote it must have been something like the spikes I have circled in red. But these values are wholly unrepresentative of the average temperature over the broader time scale and thus badly misrepresent the situation by suggesting that the 41ka warming (from the LGM baseline) is almost half as big (47%, to be precise) as the Holocene. From that graph, the LGM temperature is actually hovering around 20 (on their scale on the LHS, units of 0.1C), the Holocene is clearly below 100 on average (say 95), and the "41ka" peak is actually about 40 if we take an average over several thousand years. I've added horizontal magenta lines to indicate where I think these means lie, at least roughly. Note that they are already averaging the dust measurements over 5 points (~5000 years), and the GHG observations have a similar resolution, so using a temperature average over several thousand years would be entirely consistent and reasonable. To put it another way, if cherry-picking single data points is meaningful, then they could equally have cherry-picked the points I've circled in blue, in which case they would have concluded that the dust forcing is strongly negative, which completely destroys their claimed confidence interval in their publication.
If we (reasonably) deduce that this high-frequency variation in temperature is actually a sign of some internal (perhaps regional) variability, rather than a global response to (nonexistent) forcing changes, then the obvious thing to do is take an average over a longer interval, as I've indicated with the magenta lines. The only problem is that as soon as we do that, the need for a strong dust forcing simply melts away, and the resulting climate sensitivity estimate of 2-3.9C is entirely unremarkable (and completely consistent with what we already know). That would still be tighter than many other published estimates, but they've openly ignored several sources of uncertainty in their calculation.
There are a few other highly dubious things in the paper (such as their treatment of albedo, and ignoring changes in orbital parameters), but it's hardly worth bothering with lesser nit-picks when the temperature analysis is so clearly flawed. Thankfully, the paper seems to have been largely ignored, but I see that World Climate Report embarrassed themselves by puffing it. That's a shame as I was actually going to write something moderately nice about them (OK, I will anyway, probably over the weekend).