  Quantum geometry, which encompasses both Berry curvature and the quantum
metric, plays a key role in multi-band interacting electron systems. We study
the entanglement entropy of a region of linear size $\ell$ in fermion systems
with nontrivial quantum geometry, i.e. whose Bloch states have nontrivial $k$
dependence. We show that the entanglement entropy scales as $S = \alpha
\ell^{d-1} \ln\ell + \beta \ell^{d-1} + \cdots$ where the first term is the
well-known area-law violating term for fermions and $\beta$ contains the
leading contribution from quantum geometry. We compute this for the case of
uniform quantum geometry and cubic domains and provide numerical results for
the Su-Schrieffer-Heeger model, 2D massive Dirac cone, and 2D Chern bands. An
experimental probe of the quantum geometric entanglement entropy is proposed
using particle number fluctuations. We offer an intuitive account of the
area-law entanglement related to the spread of maximally localized Wannier
functions.
