  In this paper, we consider isotropic and stationary real Gaussian random
fields defined on $\mathbb{S}^2\times\mathbb{R}$ and we investigate the
asymptotic behavior, as $T\rightarrow +\infty$, of the empirical measure
(excursion area) in $\mathbb{S}^2\times\mathbb{R}$ at any threshold, covering
both cases when the field exhibits short and long memory, i.e. integrable and
non-integrable temporal covariance. It turns out that the limiting distribution
is not universal, depending both on the memory parameters and the threshold. In
particular, in the long memory case a form of Berry's cancellation phenomenon
occurs at zero-level, inducing phase transitions for both variance rates and
limiting laws.
