In this paper, we present several explicit reconstructions for a novel
relativistic theory of modified Newtonian dynamics (RMOND) derived from the
background of Friedmann-Lemaı^tre-Robertson-Walker cosmological
evolution. It is shown that the Einstein-Hilbert Lagrangian with a positive
cosmological constant is the only Lagrangian capable of accurately replicating
the exact expansion history of the Λ cold dark matter (ΛCDM)
universe filled solely with dust-like matter and the only way to achieve this
expansion history for the RMOND theory is to introduce additional degrees of
freedom to the matter sectors. Besides, we find that the ΛCDM-era also
can be replicated without any real matter field within the framework of the
RMOND theory and the cosmic evolution exhibited by both the power-law and
de-Sitter solutions also can be obtained