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Hi Hasret,
LEAP is certainly flexible enough to allow this. What you could do is incorporate Average Temperature as a Key Assumption (as you said), and then use scenarios to vary the temperature as you see fit. There are several branches that one could imagine this affecting: electricity demand due to cooling or heating load, hydroelectric reserves, or perhaps primary productivity of biofuel crops, to name a few.
You would have to introduce your own temperature-dependent functions to describe how these branches would be influenced, but the form of these dependencies is for you to decide.
I've built a very simple example to illustrate how you might model electricity demand for an air conditioner, using the number of cooling degree-days and a threshold temperature to indicate air conditioning load. Note that I've greatly exaggerated the change in temperature, to make the effect more prominent. You'll be able to see how, under rising temperatures, energy demand is affected by increasing our tolerance for higher indoor temperatures. Here is the download link for the file:
ftp://forums.seib.org/areas2012/AirConditioning.leap
As a follow-up (although it may be more than you're willing to do at this point), LEAP can be integrated with its sister model WEAP (Water Evaluation and Planning, http://www.weap21.org/), which will allow you to more fully explore the impact of temperature on water resources and energy. You can read about the the model integration here:
http://sei-us.org/media/SEI-Symposium-2010_Heaps_Sieber.pdf
I know that's a fair bit of information, hopefully you find it helpful,
Taylor