Postdoc with full-time appointment while doing business part-time I understand that postdocs can be too busy to even sleep, but if we put this aside - would US school allow a postdoc (full time appointment) having his/her own business in spare time? If so, what kind of regulation is applied here? Also, would the mentor be annoyed/alerted by this kind of planning? 

I guess the field of the business also matters. For example, if the business is related with research the postdoc is doing, it is probably highly restricted; while being a landlord is something completely different. 

Any thought? Thank you in advance.
We try not to give legal advice on this forum, so the legal question of whether or not (and if so how) you can have a business on the side is one that you probably need to ask a potential employer.

But I do want to comment on a separate point. As you point out, being a postdoc is a tiring phase of your life. In essence, people who want to be postdocs do so because they aspire to getting faculty positions in their next career step. I suspect that that is the same for you.

The reason why being a postdoc is so tiring is because you're competing with others for these few faculty positions: it's not necessarily that your job description requires you to work 60 hours a week, but that you feel like you need to because you see all of these other postdocs crank out 3 or 4 papers a year, and these other postdocs are your competition. So, if you are thinking of running a business on the side, then you need to be aware of the fact that that curtails your productivity compared to other postdocs, and that that in itself curtails your chances to get a faculty positions because you will have less to show on your CV.

If you know that your chances of getting a faculty position will be significantly diminished, and you still keep your side business, then you ought to ask yourself why you want to become a postdoc.