John has already given a spot on answer. Just as an addendum (which is just a bit too long as a comment), let me just add a quote from Cohen himself (from Statistical power analysis for the behavioral sciences, 1988):
The terms "small,", "medium,", and "large" are relative, not only to each other, but to the area of behavioral science or even more particularly to the specific content and research method being employed in any given investigation [...]. In the face of this relativity, there is a certain risk inherent in offering conventional operational definitions for these terms [...]. This risk is nevertheless accepted in the belief that more is to be gained than lost by supplying a common conventional frame of reference which is recommended for use only when no better basis for estimating the ES index is available. (p. 25).
The emphasis is mine. In many cases, there is a better basis, since effects are often measured with scales for which we have some prior knowledge/intuition about the meaning of the raw units and the amount of variability in the outcome.
In his famous 1994 paper ("The earth is round (p < .05)"), Cohen himself also recommended moving away from 'standardized' measures of effect and instead advocated working with raw measures of effect. Another quote:
To work constructively with "raw" regression coefficients and confidence intervals, psychologists have to start respecting the units they work with, or develop measurement units they can respect enough so that researchers in a given field or subfield can agree to use them. In this way, there can be hope that researchers' knowledge can be cumulative. (p. 1001).
Again, emphasis is mine. It is unfortunate that Cohen got his name attached to these 'canned' values, when in fact he was quite careful not to overemphasize their meaning.