How much Lower is the Temperature on the Timpanogos Summit than on Campus?

Since the environmental lapse rate varies from day to day, and also with altitude, the exact answer cannot be calculated precisely, it must be measured.  However it is often possible to calculate a good estimate by assuming that the environmental lapse rate is at its standard (nearly its average) value, which is
3.6 °F/1000 ft.

The summit of Timpanogos is at an elevation, above sea level, of 11750 ft.

The ground about the ESC is at an elevation, above sea level, of 4655 ft.

The difference is (11750 - 4655) feet = 7095 feet.

Hence the temperature difference for a standard lapse rate is (3.6 °F/1000 ft) ´ (7095 ft) = 25.5 °F.
This is often, but not always, a very good estimate.  It tends to be least accurate when there is a
strong, deep temperature inversion.  During such an inversion there are times when the temperature
can be lower on the valley floor than on the summits of Timpanogos and other nearby mountains!
The estimate tends to be best during the warmer months of the year and during the winter months
when weather conditions are unsettled.