1. Use the ideal gas equation

For an ideal gas, $PV = nRT$ where $nR$ is constant for a fixed amount of gas.

2. Given relation between pressure and temperature

The problem states: $PT^2 = \text{constant}$ So, $P \propto \frac{1}{T^2}$

Let $P = \frac{k}{T^2}$ for some constant $k$.

3. Find how volume depends on temperature

From the ideal gas law, $V = \frac{nRT}{P}$ Substitute $P = \frac{k}{T^2}$: $V = \frac{nRT}{k/T^2} = \frac{nR}{k}T^3$ Hence, $V \propto T^3$

4. Volume expansion coefficient

The coefficient of volume expansion is defined as $\beta = \frac{1}{V}\frac{dV}{dT}$

Since $V = cT^3$ for some constant $c$, we get $\frac{dV}{dT} = 3cT^2$ Therefore, $\beta = \frac{1}{cT^3}(3cT^2) = \frac{3}{T}$

5. Match with the options

Thus, the volume expansion coefficient is $\boxed{\frac{3}{T}}$ So the correct option is D.