**Abstract.**

In this talk I will show how to extend the calculus of the abstract heat flow developed by Ambrosio-Gigli-SavarÃ© to construct an abstract $q$-heat flow as the gradient flow of the $q$-Cheeger energy. A sufficient condition for mass preservation and a calculus of functionals along this flow will be given. Having this, I will show that this abstract $q$-heat flow solves the gradient flow of the Renyi entropy in the p-Wasserstein space if the Renyi entropy is displacement convex. In case $1< p < 2$, it can be shown that this gradient flow is unique which implies that $q$-heat flow and the Renyi entropy flow can be identified.