  R\'enyi holographic dark energy (RHDE) correspondence is discussed in higher
dimensional cosmology, namely Kaluza-Klein (KK) cosmology. Both interacting and
non-interacting cosmological scenario are considered here. It is found that the
non-interacting model naturally leads to the late accelerated universe, unlike
the standard Holographic Dark Energy models in 4$D$, which requires interaction
to accommodate the late-time acceleration of the universe. The interacting
model produces an accelerating universe at late time albeit failing to attain
the estimated present value of the deceleration parameter. The evolution of
different cosmologically relevant parameters have been estimated. We consider
here two diagnostic tests namely, state-finder and $Om$ diagnostics to study
the non-interacting model which is more favoured in the light of recent
cosmological observations. Classical stability of the cosmological models are
also discussed
