  The purpose of this paper is to discuss the evolution of modified holographic
dark energy with variable $G$ in non-flat Kaluza$-$Klein universe. We consider
the non-interacting and interacting scenarios of the modified holographic dark
energy with dark matter and obtain the equation of state parameter through
logarithmic approach. It turns out that the universe remains in different dark
energy eras for both cases. Further, we study the validity of the generalized
second law of thermodynamics in this scenario. We also justify that the
statefinder parameters satisfy the limit of $\Lambda$CDM model.
