  In this note we consider problems related to parabolic partial differential
equations in geodesic metric measure spaces, that are equipped with a doubling
measure and a Poincar\'e inequality. We prove a location and scale invariant
Harnack inequality for a minimizer of a variational problem related to a doubly
non-linear parabolic equation involving the p-Laplacian. Moreover, we prove the
sufficiency of the Grigor'yan--Saloff-Coste theorem for general p > 1 in
geodesic metric spaces. The approach used is strictly variational, and hence we
are able to carry out the argument in the metric setting.
