The third law does not say that "if the entropy of a system approaches a minimum, it's temperature approaches absolut zero." It says that if the temperature approaches absolute zero, the entropy does. These are logical converses.
The second law of thermodynamics says that entropy can only increase, so if the early universe had been in a state of maximum entropy, then the cosmos would have experienced its heat death immediately after being born. This contradicts the observation that the present universe contains burning stars, heat engines, and life. These observations imply that the early universe was in a very low-entropy state, which shows that its initial conditions were extremely finely tuned. The reasons for this fine-tuning are not explained by general relativity or the standard model. Adding inflation to the model does not cure this fine-tuning problem.[Penrose 2005]
These ideas are strongly counterintuitive to most people, since we picture the early universe as an undifferentiated soup of hot gas, very much like what we might imagine a heat-dead universe to be like. In what way is the early universe not equilibrated?
We observe that the cosmic microwave background radiation's spectrum is a blackbody curve, which would normally be interpreted as evidence of thermal equilibrium. However, this observation only really tells us that the matter degrees of freedom were in thermal equilibrium. The gravitational degrees of freedom were not. In standard cosmological models, which are constructed to be as simple as possible, there are no gravitational waves. Although the real universe presumably does have gravitational waves in it, they are apparently very weak. In a maximum-entropy universe, the gravitational modes would be equilibrated with the matter degrees of freedom, and they would be very strong, as in models like Misner's mixmaster universe.[Misner 1969]
Even in Newtonian mechanics, gravitating systems violate most people's intuition about entropy. If we psssssht a bunch of helium atoms into a box through an inlet valve, they will quickly reach a maximum-entropy state in which their density is nearly constant everywhere. But in an imaginary Newtonian "box" full of gravitating particles, the maximum-entropy state is one in which the particles have all glommed onto each other in a single blob. This is because of the attractive nature of the gravitational force.
Charles W. Misner, "Mixmaster Universe", Physical Review Letters 22(1969)1071. http://astrophysics.fic.uni.lodz.pl/100yrs/pdf/07/036.pdf
Roger Penrose, 2005 talk at the Isaac Newton Institute, http://www.newton.ac.uk/webseminars/pg+ws/2005/gmr/gmrw04/1107/penrose/