  We introduce a generalization structure of the su(1,1) algebra which depends
on a function of one generator of the algebra, f(H). Following the same ideas
developed to the generalized Heisenberg algebra (GHA) and to the generalized
su(2), we show that a symmetry is present in the sequence of eigenvalues of one
generator of the algebra. Then, we construct the Barut-Girardello coherent
states associated with the generalized su(1,1) algebra for a particle in a
P\"oschl-Teller potential. Furthermore, we compare the time evolution of the
uncertainty relation of the constructed coherent states with that of GHA
coherent states. The generalized su(1,1) coherent states are very localized.
