  Burgers-Kardar-Parisi-Zhang (KPZ) scaling has recently (re-) surfaced in a
variety of physical contexts, ranging from anharmonic chains to quantum systems
such as open superfluids, in which a variety of random forces may be
encountered and/or engineered. Motivated by these developments, we here provide
a generalization of the KPZ universality class to situations with long-ranged
temporal correlations in the noise, which purposefully break the Galilean
invariance that is central to the conventional KPZ solution. We compute the
phase diagram and critical exponents of the KPZ equation with $1/f$-noise
(KPZ$_{1/f}$) in spatial dimensions $1\leq d < 4$ using the dynamic
renormalization group with a frequency cutoff technique in a one-loop
truncation. Distinct features of KPZ$_{1/f}$ are: (i) a generically
scale-invariant, rough phase at high noise levels that violates
fluctuation-dissipation relations and exhibits hyperthermal statistics {\it
even in d=1}, (ii) a fine-tuned roughening transition at which the flow
fulfills an emergent thermal-like fluctuation-dissipation relation, that
separates the rough phase from (iii) a {\it massive phase} in $1< d < 4$ (in
$d=1$ the interface is always rough). We point out potential connections to
nonlinear hydrodynamics with a reduced set of conservation laws and noisy
quantum liquids.
