On a holographic dark energy model with a Nojiri-Odintsov cut-off in general relativity

In this paper we consider the models of the accelerated expanding large scale universe~(according to general relativity) containing a generalized holographic dark energy with a Nojiri - Odintsov cut - off. The second component of the darkness is assumed to be the pressureless cold dark matter according to observed symmetries of the large scale universe. Moreover, we assume specific forms of the interaction between these two components and besides the cosmographic analysis, we discuss appropriate results from $Om$ and $Om3$ analysis and organize a closer look to the models via the statefinder hierarchy analysis, too. In this way we study mainly impact of the interaction on the dynamics of the background of our universe~(within specific forms of interaction). To complete the cosmographic analysis, the present day values of the statefinder parameters $(r,s)$ and $(\omega^{\prime}_{de}, \omega_{de})$ has been estimated for all cases and the validity of the generalized second law of thermodynamics is demonstrated. Our study showed that theoretical results from considered phenomenological models are consistent with the available observational data and symmetries.