No doubt, with this sort of noise - pinpointing it could be tough. Fortunately, this doesn't seem to be a driveability or otherwise affect the longevity of the car - but doesn't mean that the noise is right or can't drive you bonkers as you are driving it.
Like you mentioned, it could be from the body of the car - loose or ill fitting body panels, exhaust heat shield touching the chassis or exhaust system, could be a damaged powertrain mount, etc.
Since you mentioned its temperature dependence and what is know on this generation - we initially steered you toward the most likely source of the noise. If those don't solve the noise - then you'll have to work your way outward to try and solve this noise.
Depending on your dealership - you might be able to get some relief from this NVH issue (Noise Vibration Harshness). Some dealerships extend that over some period of time - some as short as a year, some may go three years or more.
When I bought my 2009 Matrix - I had this weird high pitched squeaking/rubbing noise that only happened when the weather was cold and when I took off from a full stop to about 20 MPH. Was so pronouced on my long commutes to work, that I tempted to trade the car in for something else - bothered me that much. Dealership tried their best to pinpoint it - but couldn't find it. Through forums like this, I found out that it was likely the side airbag assembly in the front seat as the source. Options were to replace the seat or try to "lube" it. Dealership wouldn't cover the cost of seat replacement, just wanted to pull the airbag assembly out and lube it. I ended up replacing the seat on my dime (kept the other seat as a "spare" - fast forward almost 7 years, noise never returned!