  Based on the generalized uncertainty principle (GUP), the critical
temperature and the Helmholtz free energy of Bose-Einstein condensation (BEC)
in the relativistic ideal Bose gas are investigated. At the non-relativistic
limit and the ultra-relativistic limit, we calculate the analytical form of the
shifts of the critical temperature and the Helmholtz free energy caused by weak
quantum gravitational effects. The exact numerical results of these shifts are
obtained. Quantum gravity effects lift the critical temperature of BEC. By
measuring the shift of the critical temperature, we can constrain the
deformation parameter $\beta_0$. Furthermore, at lower densities, omitting
quantum gravitational effects may lead to a metastable state while at
sufficiently high densities, quantum gravitational effects tend to make BEC
unstable. Using the numerical methods, the stable-unstable transition
temperature is found.
