  Capacity of entanglement (CoE), an information-theoretic measure of
entanglement, defined as the variance of modular Hamiltonian, is known to
capture the deviation from the maximal entanglement. We derive an exact
expression for the average eigenstate CoE in fermionic Gaussian states as a
finite series, valid for arbitrary bi-partition of the total system. Further,
we consider the complex SYK$_2$ model in the thermodynamic limit and we obtain
a closed-form expression of average CoE. In this limit, the variance of the
average CoE becomes independent of the system size. Moreover, when the
subsystem size is half of the total system, the leading volume-law coefficient
approaches a value of $\pi^{2}/8 - 1$. We identify this as a distinguishing
feature between integrable and quantum-chaotic systems. We confirm our
analytical results by numerical computations.
