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In this paper, we present several explicit reconstructions for a novelrelativistic theory of modified Newtonian dynamics (RMOND) derived from thebackground of Friedmann-Lema$\hat{\text{\i}}$tre-Robertson-Walker cosmologicalevolution. It is shown that the Einstein-Hilbert Lagrangian with a positivecosmological constant is the only Lagrangian capable of accurately replicatingthe exact expansion history of the $\Lambda$ cold dark matter ($\Lambda$CDM)universe filled solely with dust-like matter and the only way to achieve thisexpansion history for the RMOND theory is to introduce additional degrees offreedom to the matter sectors. Besides, we find that the $\Lambda$CDM-era alsocan be replicated without any real matter field within the framework of theRMOND theory and the cosmic evolution exhibited by both the power-law andde-Sitter solutions also can be obtained.

Reconstruction of relativistic modified Newtonian dynamics for various cosmological scenarios