Gyrochronology is semi-empirical in the sense that there is some justification for the temporal dependence that you mention. There is a line of argument for the $t^{1/2}$ dependence and it can be found on pp7-8 of this pedagogical review by Jerome Bouvier.
http://arxiv.org/pdf/1307.2891v1.pdf
The basic idea is of a spherically symmetric, ionised wind that corotates with the star, held by its radial magnetic field, but which decouples from the magnetic field at some distance away from the star carrying away angular momentum. Further assuming a linear dynamo model, such that the magnetic field scales linearly with rotation rate, yields an angular momentum loss rate that is proportional to rotation rate cubed. Equating this to $I d\omega/dt$ (assuming a constant moment of inertia) leads to $\omega \propto t ^{-1/2}$ (or $P \propto t ^{1/2}$).
Some limitations of this model (and gyrochronology in general) are reviewed here (by me!).
http://arxiv.org/abs/1404.7156
For instance on the pre-main-sequence you can't assume a constant $I$; there are different ideas for how magnetic field scales with rotation rate; different ideas about magnetic topologies and different ideas for how the wind decouples from the magnetic field at large distances. All of these things mean it would be a surprise if rotation period was exactly proportional to the square root of time. The initial conditions also play a role at young ages - although the angular momentum losses lead to a convergence of rotation periods, this takes time, and is the dominant source of uncertainty (along with differential rotation), even at older ages.
So, the approach taken is to assume that rotation period can be represented by the product of a time-dependent function (usually taken to be $t^{n}$) and another function representing a mass-dependence. Observationally, the situation is that $n$ is found to be 0.52-0.57 by comparing the rotation periods of young Sun-like stars and the Sun itself.
(e.g. Mamajek & Hillenbrand 2008, ApJ, 687, 1264). But this relationship is poorly calibrated at lower masses and also for stars older than the Sun.
