I thought this was a rather fascinating question, and it turns out some researchers have agreed - enough to conduct their own studies!
In a paper from 1985, but with too wonderful a title not to note, consider And on the Seventh Day: Faculty Consulting and Supplemental Income
Obviously one will want to review this paper, as it's full of interesting observations, such as on the topic of how much this varies by field and how often/how much they are paid:
Less than 10 percent of college and university faculty employed in
fields allied with science and engineering report supplemental
earnings that represent more than one-third of their base academic
salaries. The comparable figure for faculty employed in the humanities is only 4 percent.
So certainly consulting is much more popular in some fields than others, however also consider:
Sixty to 85 percent of all faculty report receiving some income beyond
their base academic salaries. Supplemental income results from all
forms of income-generating activities (for example, research and
teaching during the summer months as well as consulting) and is earned
both within and without the institution. The amount represents only
about 15 percent of average basic academic salaries. About half of all
college and university faculty report having some form of "outside"
supplemental income during a given year.
So not only does it vary widely by field, but there is also a considerable amount of variation between individuals.
The bottom-line of the report is that a little consulting is extremely common, but consulting a lot is relatively rare - only 5-6% of faculty report consulting more than 1 day per week on average. This of course also permits variance throughout the year, with more consulting happening when classes are not in session. For non-US natives, it is important to note that in the US the concept of an "academic/9-month year" is common, and students and professors alike can take the summer off or work on their careers, take extra-paid employment (including classes), etc...and some people get stiffed and have to work for no additional pay, but that's unpleasant to think about!
This is certainly an older study, so let's consider some more recent research: Outside Consulting Income by University Faculty in Health Administration
Sadly this is pay-walled, but the abstract has good info that agrees with the older study nicely:
Based on a comprehensive survey of health administration (HA) faculty
in the US, the current study presents data on the frequency, dollar
amounts, and determinants of outside consulting income among
respondents. Approximately three quarters of respondents engage in
some consulting activities that yield, on average, approximately 25%
additional income above one's university base salary. However averages
can be misleading given that substantial variation in earnings exists
among respondents at each rank. Median consulting incomes were
approximately 9% of respondents' base salary. Various factors
including rank, gender, and professional accomplishments were
associated with engaging in any consulting activities. Among those who
consult, school of employment, gender, and self-reported expertise are
associated with the amount of consulting income earned.
In a quick Google Scholar search this question seems to have been highly studied in the 1980's, and so the foundation literature is from this time. More recent studies seem to generally accept the older studies conclusions, and then get ever more specific - examining cultural differences (like attitudes of Arab faculty and how it relates to consulting), individual fields and schools (comparing various law professors and positions in their use of consulting), etc.
If you are interested in this area I'd strongly recommend starting with a full read of "And on the Seventh Day" quoted above, as it also deals with thinks like reasons (which seem not to be strictly economic!), effect on research/teaching, and so on. Then some more specific poking around can narrow down your question, or it might just answer your question entirely!