First off, your Byetta is not magically programmed to self destruct at day 30; otherwise how could they ever store the stuff in warehouses for distribution. I routinely use EVERY BIT of Byetta in my pen and that takes me well past the 30 day mark and it is still as effective as the first day I used it.
Now, about using your pen after 77 degrees. I have never in all the time I've used Byetta (over a year) had to deal with a pen that has gotten left in the sun. I used to heat my Byetta up by putting the pen under my arm so I wasn't injecting 35 degree liquid, so its possible I heated it to over 77 degrees. The best indicator of is your Byetta still working is your Glucose meter. If you go a couple of days where your blood sugar is high and the injections don't seem to be pulling it down to where you are used to seeing it, well the proteins in the Byetta probably got fried and it isn't doing its thing anymore (which sadly means you are going to have to replace it). I've encountered this once, when I got a pen fresh from the pharmacy that had bad Byetta. From day one, it didn't move my blood sugar one inch. Fortunately I had some left over in another pen that did what it was supposed to and acted as a baseline for me to verify that the pen was bad.
So the upshot of all this is:
1) USE ALL THE BYETTA IN YOUR PEN!!!!!
2) Let your Glucose meter tell you if your Byetta is working or not.
Hope this helps