Universe means the system along with its surroundings. I have always got this statement while studying the second law; be it a thermodynamics book (Sears, Salinger) , physical chemistry book (Atkins, Paula) or any site. But really how does this happen?
Take for example, the isothermal expansion of gas; the increase in entropy of the gas(which is our system) is given as $$\Delta S_{\text{sys}} = nR\ln\dfrac{v_f}{v_i}$$
Since the the surroundings remain at constant pressure, the change in enthalpy is same as the heat energy taken by the system. Therefore the entropy of the surroundings is given by $$\Delta S_\text{surr} = -nR\ln\dfrac{v_f}{v_i}$$
Therefore the entropy if the universe is what? $0$; Then how can the entropy of the universe increase? It remains the same! Then why is the statement telling otherwise?
Best Answer
The statement "entropy of the universe increases" is a doubtful extrapolation of thermodynamics.
The problem with that statement, as CuriousOne has written in the comments, is that universe is not a closed system amenable to thermodynamic description. It has no volume, no temperature and we have no means to experiment with it the way we do with thermodynamic system such as gas in a piston. For example, heat exchange between the Universe and environment has no meaning because there is no environment.
Thus thermodynamic laws do not apply to it with the same certainty they do to lab systems and it has no meaning to ascribe it the Clausius entropy (via integral of $dQ/T$).
To make the statements in books meaningful, often it helps to replace the word "universe" with "closed system consisting of the studied system and the surrounding system it interacts with" supposing these two can attain states of thermodynamic equilibrium. Then the entropy or any other thermodynamic state quantity ascribed to these may have meaning.