  Implementing a family of differential rotation laws inspired by binary
neutron-star merger remnants, we consider the impact of the rotation profile on
the low-T/W instability. We use time evolutions of the linearised dynamical
equations, in Newtonian gravity, to study non-axisymmetric oscillations and
identify the unstable modes. The presence and evolution of the low-T/W
instability is monitored with the canonical energy and angular momentum, while
the growth time is extracted from the evolved kinetic energy. The results for
the new rotation laws highlight similarities with the commonly considered
j-constant law. The instability sets in when an oscillation mode co-rotates
with the star (i.e. whenever there is a point where the mode's pattern speed
matches the bulk angular velocity) and grows faster deep inside the co-rotation
region. However, the new profiles add features, like an additional co-rotation
point to the problem, which affect the onset of instability. The rotation laws
influence more drastically the oscillation frequencies of the l=m=2 f-mode in
fast rotating models, but affect the instability growth time at any rotation
rate. We also identify models where the low-T/W instability appears to be
triggered by inertial modes. We discuss to what extent the inferred qualitative
behaviour is likely to be of observational relevance.
