  (abridged) Context: Solar-like differential rotation is characterized by a
rapidly rotating equator and slower poles. However, theoretical models and
numerical simulations can result in a slower equator and faster poles when the
rotation is slow. Aims: We study the critical rotational influence under which
differential rotation flips from solar-like (fast equator, slow poles) to an
anti-solar one (slow equator, fast poles). We estimate the non-diffusive
($\Lambda$ effect) and diffusive (turbulent viscosity) contributions to the
Reynolds stress. Methods: We present the results of three-dimensional numerical
simulations of mildly turbulent convection in spherical wedge geometry. Here we
apply a fully compressible setup which would suffer from a prohibitive time
step constraint if the real solar luminosity was used. We regulate the
convective velocities by varying the amount of heat transported by thermal
conduction, turbulent diffusion, and resolved convection. Results: Increasing
the efficiency of resolved convection leads to a reduction of the rotational
influence on the flow and a sharp transition from solar-like to anti-solar
differential rotation for Coriolis numbers around 1.3. We confirm the recent
finding of a large-scale flow bistability: contrasted with running the models
from an initial condition with unprescribed differential rotation, the
initialization of the model with certain kind of rotation profile sustains the
solution over a wider parameter range. Conclusions: Our results may have
implications for real stars that start their lives as rapid rotators implying
solar-like rotation in the early main-sequence evolution. As they slow down,
they might be able to retain solar-like rotation for lower Coriolis numbers
before switching to anti-solar rotation. This could partially explain the
puzzling findings of anti-solar rotation profiles for models in the solar
parameter regime.
