$PV = nRT$ and $P = kV^{-2}$.

$\Rightarrow kV^{-2}V = kV^{-1} = nRT$

Differentiating, we get: $-kV^{-2} dV = nRdT \rightarrow -PdV = nRdT$

$\Rightarrow dT = \large\frac{-PdV}{nR} $$= \large\frac{-P(4V_0-V_0)}{nR} $$ = \large\frac{-nRT_0}{nRV_0}$$3V_0$ $ = -3T_0$