Solution

Assuming we're working with ideal gases
If the gas is expanding according the law
$PV^2=const$
From the ideal gas law we have
$PV=nRT$
$P=\frac{nRT}{V}$
and substituting this into the given law, we get
$TV=const$
since n and R are constant
$T=\frac{const}{V}$
so, T is inversely proportional to V, if V increases, T decreases, therefore, the gas cools down