If you are looking for a great project that answer your question, then consider the American University in Cairo new campus! The university recently moved its campus from Cairo to its outskirts. The new campus is the embodiment of Vernacular Architecture, Climate Change and Sustainability. Amir
A long while (nearly 30-40 years) ago in the zenith of modern movement in architecture when there was very little if any concern about climate change, architects such as Hassan Fathy pioneered vernacular architecture with an eye on their lower environmental impacts as well as the other two less-attended-to pillars of sustainability, i.e. social and economic. However, the notion of 'vernacular' is somewhat overrated and sometimes even abused, so we should be careful about how we utilise it. There are even known and widely researched principles on which vernacular and traditional architecture in the Middle East (where the climate conditions have always known to be harsher than most parts of the world) have been established some of which are exactly the same as what is known to form the bases for sustainable architecture/design. Again all such studies predate sustainability and climate change.
The short answer to this question is that there are positive correlations between the two for some very obvious reasons. The more detailed answer will be in line with what Ilan Kelman rightly pointed out. A simple online search will help you to find even more resources addressing this issue.