Separating my reply in two since this part is silly bonus.
Last year there was this thread and post. To summarize the post, you could get a step more fundamental by using the quantity \beta=\frac{1}{\tau} (or negative that). This has two advantages. First, equations tend to be a lot cleaner in terms of \beta. Second, it eliminates a very niche but real aspect of temperature in which negative temperatures are possible and actually hotter than positive temperatures. So, something at -1 kelvin will give up heat to something at 1 billion kelvin. Negative temperatures are not possible in “normal” systems, though, and require some sort of contrivance to realize.
All the same, \beta gets lots of play given the mathematical cleanliness it provides.