is there a second order approximation to perceived loudness
This depends highly on the signal and it's content. For narrow band signal, the loudness can fairly well be estimated through the equal loudness curves as published in ISO 226 (see for example) http://en.wikipedia.org/wiki/Equal-loudness_contour 

For wide band signal, things are more complicated. If the signals are stationary, you apply frequency weighting curves for the appropriate overall loudness (such as "A" or "C" weighting), which gives in some cases a reasonable estimate but can also be quite off by  a fair amount. 

Next step up in complexity would be a loudness calculation based on on a detailed perceptual model. So you split the signal into "critical bands", apply forward, backward, simultaneously masking thresholds, apply frequency weightings and than integrate this all up. The resulting loudness is often measure in "Sone" which intended to be a "ratio scale", meaning it has an absolute zero value (threshold of audibility) and 6 Sones is twice as loud as 3 Sone.

So overall it's possible to get very accurate estimates of perceived loudness even for non stationary signals, but it requires deploying very complicated perceptual models. These models basically the same ones as used in perceptual codecs such as MP3, AAC, Dolby Digital etc.