Using molecular dynamics (MD) simulations of a generic model, we investigated heat propagation in bottle-brush polymers (BBP). An architecture is referred to as a BBP when a linear (backbone) polymer is grafted with the side chains of different length Ns and grafting density ρg, which control the bending stiffness of a backbone. Investigating κ-behavior in BBP is of particular interest due to two competing mechanics: increased backbone stiffness, via Ns and ρg, increases the thermal transport coefficient κ, while the presence of side chains provides additional pathways for heat leakage. We show how a delicate competition between these two effects controls κ. These results reveal that going from a weakly grafting (ρg < 1) to a highly grafting (ρg ≥ 1) regime, κ changes non-monotonically that is independent of Ns. The effect of side chain mass on κ and heat flow in the BBP melts is also discussed.