  We complete our previous investigation concerning the structure and the
stability of "isothermal" spheres in general relativity. This concerns objects
that are described by a linear equation of state $P=q\epsilon$ so that the
pressure is proportional to the energy density. In the Newtonian limit $q\to
0$, this returns the classical isothermal equation of state. We consider
specifically a self-gravitating radiation (q=1/3), the core of neutron stars
(q=1/3) and a gas of baryons interacting through a vector meson field (q=1). We
study how the thermodynamical parameters scale with the size of the object and
find unusual behaviours due to the non-extensivity of the system. We compare
these scaling laws with the area scaling of the black hole entropy. We also
determine the domain of validity of these scaling laws by calculating the
critical radius above which relativistic stars described by a linear equation
of state become dynamically unstable. For photon stars, we show that the
criteria of dynamical and thermodynamical stability coincide. Considering
finite spheres, we find that the mass and entropy as a function of the central
density present damped oscillations. We give the critical value of the central
density, corresponding to the first mass peak, above which the series of
equilibria becomes unstable. Finally, we extend our results to d-dimensional
spheres. We show that the oscillations of mass versus central density disappear
above a critical dimension d_{crit}(q). For Newtonian isothermal stars (q=0) we
recover the critical dimension d_{crit}=10. For the stiffest stars (q=1) we
find d_{crit}=9 and for a self-gravitating radiation (q=1/d) we find
d_{crit}=9.96404372... very close to 10. Finally, we give analytical solutions
of relativistic isothermal spheres in 2D gravity.
