On the contrary, the science doesn’t support your hypothesis. Hair grows at a pretty steady rate, a rate that is really only affected by changes in diet. Hair is a outgrowth of protein, so anything that tunes that process, such as proper hydration and diet, will slightly increase the rate at which your hair grows, but it’s not going to go much past about 1/2 inch per month.
The exception to that rule is pregnant women, who can see upwards of 1 to 1–1/2” inches a month of hair growth.
The only thing I can imagine is that you are making more Vitamin D during the summer months, as you are getting more exposure to UV light. I don’t really know the processes all that well, but perhaps Vitamin D has something to do with hair growth or, alternately, skin elasticity and hydration. The changes in skin between the seasons would explain the prominence of your 5 o’clock shadow, at least.
And @Wizard, there is no correlation between sweat and hair growth.