Someone linked to my blog at reddit, and someone else in response linked to an interesting take by John Bryant [pdf] on using the ideal gas equation of state as the quantity theory of money. Bryant's equation is (brackets in the original):
P V = [N k] T
Where P is the GDP deflator, V is the output volume (which I think is RGDP = Y), N is the money stock (he uses M4) and T is velocity (it'd be velocity of M4; he calls it the index of trading value). That is to say, he exactly reproduces the quantity theory of money equation of exchange:
P Y = M V
since M is M4 and V is the velocity of M4: V = PY/M. Bryant then uses various thermodynamic relationships to look at changes in P, V, N and T.
P = k M^(k - 1)
Where the relationship between money and the price level is analogous to an isentropic (adiabatic) expansion log P - γ log V = constant. The analog of the ideal gas equation of state would be 
P M = k N
or in terms of the symbols of the equation of exchange
P M = k P Y
Or in terms of Bryant's symbols:
P [N k] = P V T
Bryant later gets a changing elastic index (adiabatic index, related to the information transfer index), but as yet, I haven't fully digested it.
Overall, it is a different take, but may include some interesting insights.
 If you want to make this look more like the equation of exchange, you would write it as:
P M = k N
P N^(1/k) = k M^k-1 M
N^(1/k) = M
(PY)^(1/k) = M
I've set a bunch of constants equal to 1 in the entire above presentation.