  In this work we develop the path integral optimization in a class of
inhomogeneous 2d CFTs constructed by putting an ordinary CFT on a space with a
position dependent metric. After setting up and solving the general
optimization problem, we study specific examples, including the M\"obius, SSD
and Rainbow deformed CFTs, and analyze path integral geometries and complexity
for universal classes of states in these models. We find that metrics for
optimal path integrals coincide with particular slices of $AdS_3$ geometries,
on which Einstein's equations are equivalent to the condition for minimal path
integral complexity. We also find that while leading divergences of path
integral complexity remain unchanged, constant contributions are modified in a
universal, position dependent manner. Moreover, we analyze entanglement
entropies in inhomogeneous CFTs and show that they satisfy Hill's equations,
which can be used to extract the energy density consistent with the first law
of entanglement. Our findings not only support comparisons between slices of
bulk spacetimes and circuits of path integrations, but also demonstrate that
path integral geometries and complexity serve as a powerful tool for
understanding the interesting physics of inhomogeneous systems.
