Well, not all "western" environments are cold, and not all "eastern" environments are hot. I think we are generalizing the use of eastern and western here. Greek, and rest of southern Europe isn't very cold but the Greeks were known for boxing, and that part of the world might have been the first to do so. Ancient Greeks weren't going around hearing heavy boots all the time. China, Korea, and northern Japan gets pretty damn cold yet they have kicking styles. If you look at medieval Chinese and Korean clothing, there are heavy leather boots. I just don't think climate is an accurate indicator of what kind of martial arts develop. I'm sure it has some affect since environments influence culture, but I doubt it has had a major impact.
I mispoke by not placing my statements into a historical context.
When we are talking about the origin of Karate/judo for example, we are not really talking about samurai or other historical japanese arts (even though karate and judo take inspiration from historical arts), we are talking about societal conditions in the late 19th/early 20th century.
Similarly when we talk about the origin of boxing, we're talking about living conditions in Europe (mainly Britain) and America during the 18th and early 19th century - because this is the part of the world where modern boxing developed into a sanctioned sport, kind of a ritualized form of combat. My best guess is that bunch of people started punching each other in bars and it grew into more and more of a spectacle, people started making money off of it and holding spectator events, until eventually this show became so popular it became boxing.
All TMAs are fairly modern, the vast majority having been developed within the last 100-150 years. And few TMAs are military in nature