Why is the change in entropy for a given energy input dependent on the temperature? In my thermodinamics class, we saw that 
$$dS=\frac{dQ}{T}$$
and
$$\Delta S=\int_{a}^{b}\frac{dQ}{T}$$
My question is: why for the same energy input $dQ$ the entropy increases more in lower temperatures than in higher ones, like that formula states?
 A: Let's first consider the definitions of entropy and temperature from a fundamental physical point of view
The most general rigorous definition of the entropy of a system that we use in theoretical physics can be formulated without writing down any equations, it is simply the amount of information that you would need in order to specify the exact physical state of a system, given only its macroscopic specification. So, if you have some amount of gas at some pressure in a container of some given volume, then there are a huge number of possible quantum states available that would fit the bill.
The amount of information needed to specify which one of the huge number of states the gas actually is in, would require you to specify a binary string of a length of the logarithm of the possible number of states in base 2. Entropy is conventionally defined by taking the natural logarithm and multiplying that by Boltzmann's constant, the two definitions are thus proportional to each other.
This definition applies in principle to every physical system, regardless of whether it is in thermal equilibrium. It is thus purely the amount of information contained in the system that you have no knowledge of. 
The definition of temperature is far less general than the definition of entropy, it only applies to systems in thermal equilibrium. For an isolated system, this means that all the physical states that the system can be in, are equally likely. The notion of temperature is then made rigorous by defining it in such a way that it will correspond to the familiar notion of hot and cold, i.e. that energy wil flow spontaneously from hot to cold and not in the opposite direction.
The explaination of why heat should only flow in one direction is easy. If you have two isolated objects and bring them into cotnact with each other, then the number of states the combined system can be in may increase if energy flows from one system to another. Since in termal equilibrium all states are equally likely, you would then expect this to happen spontaneously until the most probable situation is reached.
Now, this is used to define temperature in terms of entropy, I won't repeat that argument here. What we need to answer the question is to note that we have defined temperature such that heat flow that happens because entropy increases. This in turn happens because all states are equally likely, if the system is initially restricted to a subset of all avaialble states and suddenly all states come into play then the probability distribution will become more uniform, hence the entropy (which is defined as lack of knowledge about which state the system actually is in), will increase.
The entropy increase is the sum of the entropy increases of the two systems (entropy is additative because if you need X amount of information to specify the state of one system and Y to specify the state of the other, then you need X + Y to specify them both). Then we assume that system has a lower temperature and the other has a higher temperature. The fact that total entropy is indeed going to increase then implies that when adding the same amount of heat to the system at lower temperature will lead to a higher increase than adding that heat to a system at a higher temperature. The point is then that temperature was necessarily defined in that way so as to correspond with the intuitive notions of hot and cold. 
A: I may give you an intuitive example i read before 
Imagine shouting in a street full of noise and shouting by the same amount in library 
although the shouting ( dQ ) is the same in both cases it will have a greater effect in the library ( systems with lower temperature.
i hope it helps i know only that example and not expert with Thermodynamics.
A: An intuitive explanation is that entropy is a measure of disorder, so if heat is injected into two exactly identical systems except that one is held at a higher temperature, then you can imagine that the higher temperature system's disorder will increase less because it was more disordered to start with. In other words, its harder to increase the disorder in a more disordered system.
