  We address a mean-field zero-temperature Ginzburg-Landau, or \phi^4, model
subjected to quenched additive noise, which has been used recently as a
framework for analyzing collective effects induced by diversity. We first make
use of a self-consistent theory to calculate the phase diagram of the system,
predicting the onset of an order-disorder critical transition at a critical
value {\sigma}c of the quenched noise intensity \sigma, with critical exponents
that follow Landau theory of thermal phase transitions. We subsequently perform
a numerical integration of the system's dynamical variables in order to compare
the analytical results (valid in the thermodynamic limit and associated to the
ground state of the global Lyapunov potential) with the stationary state of the
(finite size) system. In the region of the parameter space where metastability
is absent (and therefore the stationary state coincide with the ground state of
the Lyapunov potential), a finite-size scaling analysis of the order parameter
fluctuations suggests that the magnetic susceptibility diverges quadratically
in the vicinity of the transition, what constitutes a violation of the
fluctuation-dissipation relation. We derive an effective Hamiltonian and
accordingly argue that its functional form does not allow to straightforwardly
relate the order parameter fluctuations to the linear response of the system,
at odds with equilibrium theory. In the region of the parameter space where the
system is susceptible to have a large number of metastable states (and
therefore the stationary state does not necessarily correspond to the ground
state of the global Lyapunov potential), we numerically find a phase diagram
that strongly depends on the initial conditions of the dynamical variables.
