Chemistry - Dependence of change in entropy on temperature
An analogy that I like is one paraphrased from Peter Atkins' great book "The Laws of Thermodynamics - a VSI":
Imagine two rather different places in terms of noise - a silent library and a busy train station. Imagine that the temperature is the amount of noise at the given place - a measure of the "state of noise". Let's say that you laugh out loud, corresponding to supplying energy as heat.
In the library, where there is little background noise (low $T$) this would cause a large amount of disturbance and disorder - corresponding to a large increase in entropy. On the busy train station, on the other hand, you laughing would go largely unnoticed - there is lots of background noise already (high $T$). The (relative) amount of additional disturbance and disorder would be much smaller, corresponding to a smaller increase of entropy.
In a very cold system, even a tiny amount of supplied energy in terms of heat would cause a large change in entropy. On the other hand, supplying the same amount of energy to a really hot system - with lots of molecular motion going on - would cause a minuscule change in entropy.
Disclaimer: Note that the analogy and explanation above are two ways of convincing yourself that the entropy change due to added heat is inversely proportional to the system temperature. They do not claim to be correct or physical in any other sense, so read them with a (large) pinch of salt.