  Quantum information theoretical measures are useful tools for characterizing
quantum dynamical phases. However, employing them to study excited states of
random spin systems is a challenging problem. Here, we report results for the
entanglement entropy (EE) scaling of excited eigenstates of random XX
antiferromagnetic spin chains with long-range (LR) interactions decaying as a
power law with distance with exponent $\alpha$. To this end, we extend the
real-space renormalization group technique for excited states (RSRG-X) to solve
this problem with LR interaction. For comparison, we perform numerical exact
diagonalization (ED) calculations. From the distribution of energy level
spacings, as obtained by ED for up to $N\sim 18$ spins, we find indications of
a delocalization transition at $\alpha_c \approx 1$ in the middle of the energy
spectrum. With RSRG-X and ED, we show that for $\alpha>\alpha^*$ the
entanglement entropy (EE) of excited eigenstates retains a logarithmic
divergence similar to the one observed for the ground state of the same model,
while for $\alpha<\alpha^*$ EE displays an algebraic growth with the subsystem
size $l$, $S_l\sim l^{\beta}$, with $0<\beta<1$. We find that $\alpha^* \approx
1$ coincides with the delocalization transition $\alpha_c$ in the middle of the
many-body spectrum. An interpretation of these results based on the structure
of the RG rules is proposed, which is due to {\it rainbow} proliferation for
very long-range interactions $\alpha\ll 1$. We also investigate the effective
temperature dependence of the EE allowing us to study the half-chain
entanglement entropy of eigenstates at different energy densities, where we
find that the crossover in EE occurs at $\alpha^* < 1$.
