We study a two-dimensional Ising system under Glauber kinetics with an extrinsic energy barrier, submitted to asymptotically slow continuous cooling. The system will freeze into a nonequilibrium state if an appropriately defined effective time for the cooling schedule does not diverge. Starting in equilibrium beneath the critical temperature Tc, the system freezes into small clusters of flipped spins, with the energy density related to the cooling rate by a power law (for most cooling programs). Cluster-dynamical calculations and Monte Carlo simulations show that, for exponential cooling, the freezing exponent approaches a one-cluster value, which depends upon the energy barrier and the lattice type, via an intermediate regime with a higher effective exponent. Starting in equilibrium above Tc, the frozen state consists of large domains of either phase. Simple interface-dynamical arguments suggest that the frozen correlation function should assume a scaling form, with a (universal) scaling exponent, which is the same as for domain growth after an instantaneous quench. Monte Carlo simulations find evidence for this scaling form at small values of the scaling variable only, suggesting the importance of initial correlations for a very wide regime of cooling rates. In neither case does the temporal evolution of the frozen state follow a Kohlrausch form, suggesting a qualitative distinction from true glassy states.