The topological metal states in electronic systems have been extensively studied, but topological phonons were explored only in a few examples so far. Here, we expose that the topological type-I and type-II Weyl phonons simultaneously appear in a recently realized new carbon allotrope, $T$-carbon, and the topological surface states of phonons are well separated from its bulk phonon spectra that are suitable for experiments. At about 15.2 THz, we find that there exist three mutually orthogonally intersecting nodal loops (named nodal gimbal phonons) centered at the $\Gamma$ point, and two pairs of type-I Weyl phonons on the boundary of Brillouin zone around each $X$ point. In addition, there exist three pairs of type-II Weyl phonons at about 14.5 THz around each $L$ point. It is shown that these exotic topological phonons are protected by corresponding symmetries, and lead to topologically nontrivial surface states. Our findings not only afford plenty of intriguing topological phonon states in a simple material like $T$-carbon but also provide a platform to study novel properties of topological phonons, which would facilitate further both experimental and theoretical works in future.