  We study the thermodynamic properties of a rigidly rotating relativistic Bose
gas. First, we derive the solution of the equation of motion corresponding to a
free rotating complex Klein-Gordon field and determine the free propagator of
this model utilizing the Fock-Schwinger proper-time method. Using this
propagator, we then obtain the thermodynamic potential of this model in the
zeroth and first perturbative level. In addition, we compute the
nonperturbative ring contribution to this potential. Our focus is on the
dependence of these expressions on the angular velocity, which effectively acts
as a chemical potential. Using this thermodynamic potential, we calculate
several quantities, including the pressure, angular momentum and entropy
densities, heat capacity, speed of sound, and moment of inertia of this rigidly
rotating Bose gas as functions of temperature ($T$), angular velocity
($\Omega$), and the coupling constant ($\alpha$). We show that certain
thermodynamic instabilities appear at high temperatures and large couplings.
They are manifested as zero and negative values of the above quantities,
particularly the moment of inertia and heat capacity. Zero moment of inertia
leads to the phenomenon of supervorticity at certain $T$ or $\alpha$.
Supervortical temperatures (couplings) decrease with increasing coupling
(temperature). We also observe superluminal sound velocities at high $T$ and
for large $\alpha$.
