I have Skype interviewed the best 3 applicants for a PhD position, offered with my startup package as assistant professor in a USA university (ranked between 30 and 70 nationally). The candidates are all pretty much there. I am slightly leaning toward a candidate, just because he is doing an internship in the lab of a professor whom I know and who carries on research in a field very close to mine. The internship lasts 6 month, and he only did 3 months there. The candidate already provided the 3 requested letters of recommendation, but none of those is from the above-mentioned professor, probably because he has only been there for a short time. Note that I do not personally know this professor, do you think it is ok if I contact him by email to provide me an honest impression he got of the candidate?
I don't know about HR standards for PhD candidates versus faculty positions. However, I have chaired faculty search committees and the general practice for equal opportunity hiring is that you treat all of the candidates the same. So, going above and beyond for one candidate over the others could be viewed as unfair.
If that is not really an issue for you in this case, I agree with The Fire Guy's comment that the student has a reason for not including this professor as a reference. Since you do not know this professor personally, there could be a number of issues at the workplace that could lead to an unfair assessment of the candidate. Although it could be that the candidate hasn't worked with that professor long, like you suspect, it could also be that there are personality differences between the two or the professor may be an unreasonable supervisor. Also, the candidate might feel that their applying to work with you may cause problems at their current position, if this professor knows about it. If you feel that the professor's feedback is important, you could ask the candidate if he or she is okay with you contacting the professor before hand.
When I have applied for jobs, I would have been put off if I learned that the employer was contacting people who were not on my reference list. How would the candidate know that this professor is the only one you contacted?