  The covariant understanding of dispersion relations as level sets of Hamilton
functions on phase space enables us to derive the most general dispersion
relation compatible with homogeneous and isotropic spacetimes. We use this
concept to present a Planck-scale deformation of the Hamiltonian of a particle
in Friedman-Lema\^itre-Robertson-Walker (FLRW) geometry that is locally
identical to the $\kappa$-Poincar\'e dispersion relation, in the same way as
the dispersion relation of point particles in general relativity is locally
identical to the one valid in special relativity. Studying the motion of
particles subject to such Hamiltonian we derive the redshift and lateshift as
observable consequences of the Planck-scale deformed FLRW universe.
