  We develop a general theory for discontinuous non-equilibrium phase
transitions into an absorbing state in the presence of temporal disorder. We
focus in two paradigmatic models for discontinuous transitions: the quadratic
contact process (in which activation is only spread when two nearest- neighbor
sites are both active) and the contact process with long-range interactions.
Using simple stability arguments (supported by Monte Carlo simulations), we
show that temporal disorder does not destroy the discontinuous transition in
the former model. For the latter one, the first-order transition is turned into
a continuous one only in the strong-disorder limit, with critical behavior
belonging to the infinite-noise universality class of the contact process
model. Finally, we have found that rare temporal fluctuations dramatically
changes the behavior of metastable phase turning it into a temporal Griffiths
inactive phase characterized by an exponentially large decay time.
