Paul Krugman makes this observation, which I believe dates back to Rortybomb and perhaps earlier. Rortybomb, Krugman, and others have cited it as evidence for demand-based, non-structural theories of unemployment. I believe this is not exactly the correct inference.
Let's say you have the iPad sector, where unemployment is one percent (990 of 1000 are employed), and the typewriter sector, where unemployment is twenty percent (800 of 1000 are employed). AD falls across the board, and that means there is in effect a proportional real wage mark-up, relative to what conditions justify, on both sectors. Let's say that in both sectors real wages are suddenly five percent too high, again because of the whack to AD, which you can view as insufficient inflationary pressure.
To continue with this deliberately artificial example, let's also say we have equal and unitary labor demand elasticity in each sector. So if the real wage is five percent too high because of the deflation, the quantity of labor demanded should fall by five percent.
In the iPad sector, employment falls five percent, or 49.5 workers, I will call it fifty even, to 940 workers employed, or the rate of unemployment goes up from one percent to six percent.
In the typewriter sector, employment falls by five percent, which leads to 760 employed out of 1000, or the rate of unemployment rises from twenty percent to twenty-four percent.
The two rates of unemployment, in the two sectors, do not go up proportionally or even close to proportionally. It is harder to boost up the unemployment rate in the typewriter sector; the intuition is simple, namely that it is easier to double the unemployment rate when that rate is starting at the very low figure of one percent. You can set this problem up in many different ways, but it's hard to avoid that core intuition.
If the rate of unemployment is going up proportionately in both sectors, it means the downturn is administering an especially strong shock to the weaker sectors. It is a "final death blow tip them over the precipice kind of shock." It does not signify AD irrelevance, but rather that negative AD shocks and preexisting structural problems somehow magnify each other.
Yet numerous times I have seen proportional changes in unemployment rates cited as evidence for a pure AD approach. That does not follow.
As a final point, please note that not all non-AD theories imply excess demand for labor in a significant number of sectors. There are plenty of non-AD theories, such as an increase in the risk premium, where there is a big negative employment move pretty much across the board. I continue to see this point misunderstood.
The Keynesian bloggers are correct to argue against AD denialists, but when it comes to their arsenal of counters they are overreaching.