  Rossby wave instabilities (RWIs) usually lead to nonaxisymmetric vortices in
protoplanetary discs and some observed sub-structures of these discs can be
well explained by RWIs. We explore how the cooling influences the growth rate
of unstable RWI modes in terms of the linear perturbation analysis. The cooling
associated with the energy equation is treated in two different ways. The first
one we adopt is a simple cooling law. The perturbed thermal state relaxes to
the initial thermal state on a prescribed cooling timescale. In the second, we
treat the cooling as a thermal diffusion process. The difference in the growth
rate between the adiabatic and isothermal modes becomes more pronounced for
discs with smaller sound speed. For the simple cooling law, the growth rates of
unstable modes monotonically decrease with the shorter cooling timescale in
barotropic discs. But the dependence of growth rate with the cooling timescale
becomes non-monotonic in non-baratopic discs. The RWI might even be enhanced in
non-barotropic discs during the transition from the adiabatic state to the
isothermal state. When the cooling is treated as the thermal diffusion, even in
barotropic disc, the variation of growth rate with thermal diffusivity becomes
non-monotonic. Further more, a maximum growth rate may appear with an
appropriate value of thermal diffusivity. The angular momentum flux (AMF) is
investigated to understand the angular momentum transport by RWI with cooling.
