Here's what physics has to say about it:
The frequency of a wave on a string is proportional to the velocity, which is:
velocity= SQRT( Tension * Length / mass)
So, if you increase the mass of the string, you must also increase the tension to keep the velocity (and therefore frequency) the same. If we are correct in assuming that flatwounds have more mass for the same gauge, then I think we have to conclude that the tension needs to be higher, but like others have said, it may vary by brand and construction (eg. A hex core may provide enough airspace to cancel out the "filled in" outer surface of a flatwound).