Are the thermodynamic analogies useful?

The short answer is: I have no idea. It is fun to think about. Moving along the demand curve is analogous to an isothermal process and there is an additional law for an isoentropic process:

$$
P (Q^s)^{1\mp1/\kappa} = \text{constant}
$$

One interesting idea is that if $\kappa \sim 1$, (as might be inferred from the price elasticities of supply/demand), then an isoentropic process could obey

$$
P (Q^s)^{1 - 1/\kappa} \simeq P (Q^s)^{0} = P  = \text{constant}
$$

So that whenever you are experiencing sticky prices, maybe the market is undergoing an isoentropic process where the number of microstates that are consistent with the given macrostate is constant. This doesn't mean supply and demand don't change; it just means that there are the same number of microstates that describe the earlier market and the later market. In thermodynamics, an isoentropic process is also reversible. Both signs are allowed in principle and in the thermodynamics case the "plus" gives you an exponent of 5/3.

PS In an ideal gas in two dimensions, $\kappa = 2 \cdot 1/2 = 1$ where there are two degrees of freedom and the 1/2 comes from $\langle x^2 \rangle / \langle 1 \rangle$ with a Gaussian distribution. However, again, the plus sign is what gives the result that agrees with experiment.