  In this manuscript, we have researched the cosmic expansion phenomenon in
flat FRW Universe through the interaction of the recently proposed
R$\grave{e}$nyi holographic dark energy (RHDE). For this reason, we assumed
Hubble (H) and Granda--Oliveros (GO) horizons as IR cut-off in the framework of
$f(R, T)$ gravity. With this choice for IR cut-off, we can obtain some
important cosmological quantities such as the equation of state $\omega_{_T}$,
energy density $\rho_{_T}$, density parameter $\Omega_{_T}$, and pressure
$p_{_T}$, which are the function of the redshift $z$. It is observed that in
both IR cut-offs the EoS parameter displays quintom-like behaviour for three
different values of $\delta$. Here we plot these parameters versus redshift $z$
and discuss the consistency of the recent findings. Next, we explore the
$\omega_{_T}$-$\omega_{_T}^{'}$ plane and the stability analysis of the dark
energy model by a perturbation method. Our findings demonstrate that the
Universe is an accelerating model of rapid growth that is explained by quintom
like behaviour.Hence the feasibility of the RHDE model with Hubble and GO
cut-off is supported by our model. The results indicate that the IR cut-offs
play a significant role in the understanding of the dynamics of the universe.
