> If, for example, your universe is entropically uniform, is there an "arrow of time"?
No. For a system at thermodynamics equilibrium entropy is constant, neither increasing nor decreasing. So there is no arrow of time, no sense in which one direction is "the future" and the other "the past".
If a universe were entropically uniform, wouldn't the arrow of time definitely be negative? Wouldn't any random evolution result in a lessening of entropy by definition?