  We determine the quasistationary distribution of Floquet-state occupation
probabilities for a parametrically driven harmonic oscillator coupled to a
thermal bath. Since the system exhibits detailed balance, and the canonical
representatives of its quasienergies are equidistant, these probabilities are
given by a geometrical Boltzmann distribution, but its quasitemperature differs
from the actual temperature of the bath, being affected by the functional form
of the latter's spectral density. We provide two examples of quasithermal
engineering, i.e., of deliberate manipulation of the quasistationary
distribution by suitable design of the spectral density: We show that the
driven system can effectively be made colder than the undriven one, and
demonstrate that quasithermal instability can occur even when the system is
mechanically stable.
