A universe infinite in age violates the second law of thermodynamics. The second law requires entropy to increase as time goes on. For an infinite universe, entropy would have to approach its maximal value. A universe infinite in extent does not change the picture because you can look at the entropy for specfic volumes, or specific masses, and still get the same result.

An infinite time would allow the universe to reach maximal disorder. Since this is obviously not the case, the universe cannot have an infinite age.