  We study noncommutative deformations of the wave equation in curved
backgrounds and discuss the modification of the dispersion relations due to
noncommutativity combined with curvature of spacetime. Our noncommutative
differential geometry approach is based on Drinfeld twist deformation, and can
be implemented for any twist and any curved background. We discuss in detail
the Jordanian twist $-$giving $\kappa$-Minkowski spacetime in flat space$-$ in
the presence of a Friedman-Lema\^{i}tre-Robertson-Walker (FLRW) cosmological
background. We obtain a new expression for the variation of the speed of light,
depending linearly on the ratio $E_{ph}/E_{LV}$ (photon energy / Lorentz
violation scale), but also linearly on the cosmological time, the Hubble
parameter and inversely proportional to the scale factor.
