1. Write the structures of the acids

   • Peroxodisulphuric acid: $\mathrm{H_2S_2O_8}$
   • Pyrosulphuric acid: $\mathrm{H_2S_2O_7}$

2. Count $\pi$-bonds in peroxodisulphuric acid, $\mathrm{H_2S_2O_8}$

   Its structure can be represented as: $\mathrm{HO{-}S(=O)_2{-}O{-}O{-}S(=O)_2{-}OH}$

   • Each sulphur atom has two $\mathrm{S=O}$ double bonds.
   • Total number of $\mathrm{S=O}$ double bonds: $2 + 2 = 4$
   • Each double bond contributes one $\pi$-bond.

   Therefore, total $\pi$-bonds in $\mathrm{H_2S_2O_8}$: $4$

3. Count $\pi$-bonds in pyrosulphuric acid, $\mathrm{H_2S_2O_7}$

   Its structure can be represented as: $\mathrm{HO{-}S(=O)_2{-}O{-}S(=O)_2{-}OH}$

   • Again, each sulphur atom has two $\mathrm{S=O}$ double bonds.
   • Total number of $\mathrm{S=O}$ double bonds: $2 + 2 = 4$
   • Hence total $\pi$-bonds: $4$

4. Sum of $\pi$-bonds

   $4 + 4 = 8$

Therefore, the required sum is: $\boxed{8}$