Non-abelian symmetries play a central role in many areas in physics, and have been recently argued to result in distinct quantum dynamics and thermalization. Here we unveil the effect that the non-abelian SU(2) symmetry, and the rich Hilbert space structure that it generates for spin 1/2 systems, has on the average entanglement entropy of random pure states and of highly-excited Hamiltonian eigenstates. Focusing on the zero magnetization sector (J_z=0) for different fixed spin J, we show that the entanglement entropy has a leading volume law term whose coefficient s_A depends on the spin density j=2J/L, with s_A(j --> 0)=ln(2) and s_A(j --> 1)=0. We also discuss the behavior of the first subleading corrections.