A solid has a volume $v_0$ when external pressure is  $p_0$. Bulk modulus of the material is $B$. Now the pressure is increased to $p$. The additional energy per unit volume now stored in the material is $\frac{3(p^2-p_0^2)}{zB}$. Find $z=?$

For any small change of pressure dp, there will be a change of volume dV and  $\text{dp=-B}\frac{\text{dV}}{\text{V}}$ . 

In this change, work is done on the system and the energy stored in the material is
$\text{dW=-pdV}\left(\frac{\text{V}}{\text{B}}\right)\text{pdp}$ 
In the change mentioned in the question, the total work done is
$\text{W=-}{\int }_{\text{v}}^{\text{v}}\text{pdV}\text{=}{\int }_{\text{v}}^{\text{v}}\frac{\text{V}}{\text{B}}\text{pdp}$ 
The change in volume is negligible and volume can be treated as constant.
 $\text{W=}\frac{{\text{V}}_{\text{0}}}{\text{B}}{\int }_{{\text{p}}_{\text{0}}}^{\text{p}}\text{pdp}\text{=}\frac{{\text{V}}_{\text{0}}}{\text{B}}\left(\frac{{\text{p}}^{\text{2}}{\text{-p}}_{\text{0}}^{\text{2}}}{\text{2}}\right)$
Extra energy stored per unit volume is
$\frac{\text{W}}{{\text{V}}_{\text{0}}}\text{=}\frac{\text{1}}{\text{2B}}\left({\text{p}}^{\text{2}}{\text{-p}}_{\text{0}}^{\text{2}}\right)$