I've done a search & found this:
https://www.wunderground.com/resources/climate/strato_cooling.aspGreenhouse gases also cause stratospheric coolingHowever, this recovery of the ozone layer is being delayed. A significant portion of the observed stratospheric cooling is also due to human-emitted greenhouse gases like carbon dioxide and methane. Climate models predict that if greenhouse gases are to blame for heating at the surface, compensating cooling must occur in the upper atmosphere. We need only look as far as our sister planet, Venus, to see the truth of this theory. Venus's atmosphere is 96.5% carbon dioxide, which has triggered a run-away greenhouse effect of truly hellish proportions. The average surface temperature on Venus is a very toasty 894 °F! However, Venus's upper atmosphere is a much colder than Earth's upper atmosphere. The explanation of this greenhouse gas-caused surface heating and
upper air cooling is not simple, but good discussions can be found at Max Planck Institute for Chemistry and realclimate.org for those unafraid of radiative transfer theory. One way to think about the problem is that the amount of infrared heat energy radiated out to space by a planet is roughly equal to the amount of solar energy it receives from the sun. If the surface atmosphere warms, there must be compensating cooling elsewhere in the atmosphere in order to keep the amount of heat given off by the planet the same. As emissions of greenhouse gases continue to rise, their cooling effect on the stratosphere will increase. This will make recovery of the stratospheric ozone layer much slower.
Greenhouse gases cause cooling higher up, too
Greenhouse gases have also led to the cooling of the atmosphere at levels higher than the stratosphere. Over the past 30 years, the Earth's surface temperature has increased 0.2-0.4 °C, while the temperature in the mesosphere, about 50-80 km above ground, has cooled 5-10 °C (Beig et al., 2006). There is no appreciable cooling due to ozone destruction at these altitudes, so nearly all of this dramatic cooling is due to the addition of greenhouse gases to the atmosphere. Even greater cooling of 17 °C per decade has been observed high in the ionosphere, at 350 km altitude. This has affected the orbits of orbiting satellites, due to decreased drag, since the upper atmosphere has shrunk and moved closer to the surface (Lastovicka et al., 2006). The density of the air has declined 2-3% per decade the past 30 years at 350 km altitude. So, in a sense, the sky IS falling!