What is a thermal distribution of eccentricities?
The thermal eccentricity distribution was first calculated by Jeans 1919. The probability density function is indeed $$f(e) = 2e.$$

See this blog post for a nice derivation of this beautiful result, which is independent of the "temperature". 

The derivation relies on a small 'swindle' by assuming a population of only binaries and not any single or trinary systems; it also has divergent energy since it allows too many arbitrarily tight or loose binaries - it can only be an approximation to real systems.