I live in AZ! Is it possible that my Latisse bottle wont work if it was in the mailbox in warm a temperature for approximately 1

This should not impact the effectiveness of your Latisse. Keep in mind, Latisse results may plateau. A combination approach is needed. In combination with Latisse, I typically inject and apply topically PRP for a few treatments at the beginning to improve the growth rate of the lashes. Take viviscal as well. See an expert for a formal evaluation. Best, Dr. Emer

lStudies to date would suggest that Latisse has remarkable "thermal stability" - at least up to 50 degrees Clecius.
A 2011 study by Johnson and colleagues looked at how the concentration of Latisse (and other prostaglandins) changed when exposed to elevated temperatures including 37 C and 50 C for 3, 6, 9,15 or 30 days. By carefully assaying the concentration of Latisse at each time point the researchers found that there was no change in the concentration of Latisse over time at any of these temperatures. 
 
Is Latisse safe in my hot mail box ?
To answer this specific question, we really need to know how hot the Latisse was in the mailbox. Of course if the Latisse reaches temperatures well above 50 C we enter a grey area and don't fully know the answer based on all studies to date. However Latisse can withstand quite a bit of heat due to its remarkable thermable stability. It is likely to be unaffected to any significant degree following a short period of time at elevated temperature.
 
Reference 
Thermal stability of bimatoprost, latanoprost, and travoprost under simulated daily use.
Johnson TV, et al. J Ocul Pharmacol Ther. 2011.

Living in Arizona, this is a common question.  We often get asked, if I leave it in my car, is it still safe to use?  The short answer is most likely yes.  In 2011, Johnson and colleagues looked at how the concentration of Latisse (and other prostaglandins) changed when exposed to elevated temperatures up to 50C or 122F.  The researchers found that there was no change in the concentration of Latisse over time at any of these temperatures.  So, assuming the temperature inside your mailbox (of the mail delivery truck) was less than 122F, it should be safe and effective to use.