"Stratospheric temperatures" refers to extremely high temperatures in the Earth's stratosphere, which is the second layer of the atmosphere above the surface. Full definition
The source of the difference in mean lapse rate feedback between the two studies is unclear, but may relate to inappropriate inclusion of stratospheric temperature response in some feedback analyses (Soden and Held, 2006). (ipcc.ch)
Although the precise causes of such differences are unclear, model biases in lower stratospheric temperature trends are likely to be reduced by more realistic treatment of stratospheric ozone depletion and volcanic aerosol forcing. (sindark.com)
, the basic issue is that stratospheric temperatures change in response to local effects, they do not change because the troposphere does (i.e. troposphere warming does NOT imply stratosphere cooling). (realclimate.org)