  Applying the cosmological principle to Finsler spacetimes, we identify the
Lie Algebra of symmetry generators of spatially homogeneous and isotropic
Finsler geometries, thus generalising Friedmann-Lema\^{i}tre-Robertson-Walker
geometry. In particular, we find the most general spatially homogeneous and
isotropic Berwald spacetimes, which are Finsler spacetimes that can be regarded
as closest to pseudo-Riemannian geometry. They are defined by a Finsler
Lagrangian built from a zero-homogeneous function on the tangent bundle, which
encodes the velocity dependence of the Finsler Lagrangian in a very specific
way. The obtained cosmological Berwald geometries are candidates for the
description of the geometry of the universe, when they are obtained as
solutions from a Finsler gravity equation.
