The result actually holds with $C=d/2$ on any compact Riemannian manifold with a non-negative Ricci curvature. This can be seen by integrating the Li-Yau inequality: Theorem 1.1 in 

[On the parabolic kernel of the Schrödinger operator][1] 


  [1]: https://projecteuclid.org/euclid.acta/1485890415