Hodge star operator and heat propagator

Question

I am currently studying the Laplacian on a Riemaniann Manifold: An introdcution to analysis on manifolds by S. Rosenberg. I am solving some of the exercises and one of them (ex.3, ch 4.1 page 113) is to prove that: $$ \star e^{-t\Delta^q}=e^{-t\Delta^{n-q}}\star $$ Where $\star$ is the Hodge star operator and and $\Delta^p=\delta d+d\delta$ is the Hodge Laplacian. I am not able to solve this one and I am not really sure how to start. Also taking into acount the Hodge theorem ($Ker \Delta^p\cong H_{dR}^p$) and De rham's theorem $H_{dR}^n\cong H_{sing}^p$ does this property of the p Laplacian and Hodge star operator I am curious if there is any relationship with Poincare duality. Any help or refrences about this would be grealty appriciated.