The equipartition theorem details that thermal energy is divided down by degrees of freedom. In higher dimensional space there are different ratios of translations to rotations, so observations of the triple state diagrams of diatomic vs monatomic molecules would diverge from observation.
The time it takes spheres to settle after falling into a square container is a function of the dimensionality of the container. Observations as we scale real experiments with steel ball bearings suggest that the balls can't move in any other dimensions.
In higher dimensional space the great geodesics are longer, we would expect a deviation in the relativistic corrections to the GPS system that we don't observe.
The interesting question however is at the largest scales. The universe might have much higher dimensions, we just happen to be in a relatively "compact" part of it...