Solution:

Given that the gas obeys the law \( VP^2 = \text{constant} \).

1. Initially:
\[
VP^2 = k
\]

2. When the volume changes from \( V \) to \( 2V \):
\[
(2V)P'^2 = k
\]
where \( P' \) is the new pressure.

Since \( VP^2 = (2V)P'^2 \), we can relate the pressures as:
\[
P'^2 = \frac{P^2}{2}
\]
\[
P' = \frac{P}{\sqrt{2}}
\]

3. Use the ideal gas law initially and finally:
\[
PV = nRT
\]
\[
P' \cdot 2V = nRT'
\]

Substitute \( P' = \frac{P}{\sqrt{2}} \):
\[
\frac{P}{\sqrt{2}} \cdot 2V = nRT'
\]

4. Simplify:
\[
\sqrt{2} PV = nRT'
\]

Since \( PV = nRT \):
\[
\sqrt{2} \cdot nRT = nRT'
\]
\[
T' = \sqrt{2} T
\]

So, the new temperature is \( \sqrt{2} T \).