Probably because so many old flutes have cracked (see The New Improved Tuning Slide for the reasons why), many are unnecessarily concerned about the care needed with wooden flutes. One issue is the effect of heat and cold. Will dashing through the snow from car to house clutching your flute mean the end? If you leave the flute in the car for a day in summer will it turn into matchwood? This paper records an experiment to illustrate these fears are groundless.
To test the effect of cold on an instrument, I popped a spare flute head into the freezer compartment of our domestic refrigerator. I also included a thermometer, which soon registered around -20 degrees Celsius (about -4F). The flute head soon frosted over with ice. I left it there for several days, then took it out, put it on the flute and started playing.
I expected that I would not be able to play for long, thinking the extreme cold might make it painful. Wood is such a good insulator however that the sensation of cold was mild and short-lived. The next thing I expected was that severe condensation would form, but again this did not happen - condensation was not much more than normal. I didn't expect anything ghastly to happen to the head itself, and indeed it escaped unscathed.
So, several days at well-below-freezing temperatures followed by immediate playing couldn't hurt a head. What about the other extreme? I put the same head into our domestic oven, set for 50 degrees Celsius (122F) and fan-forced. Again the hapless thermometer went in as well to check the temperature.
I only left the head in there for a bit over a day, as I was interested only in testing the effect of temperature. The hot dry air in the oven would, if you left the head for much longer, dry the moisture out of the wood in the flute, emulating exposing the flute to low humidity situations. That is another experiment currently underway.
Sufficient to say, nothing horrible happened to the flute. Again, I played it immediately upon taking it out, with no apparent effect.
We can safely conclude that exposure of wooden flutes to the normal extremes of temperature likely to be experienced are unlikely to affect them.