We study topological modes in relativistic hydrodynamics by weakly breaking the conservation of energy momentum tensor. Several systems have been found to have topologically nontrivial crossing nodes in the spectrum of hydrodynamic modes and some of them are only topologically nontrivial with the protection of reflection symmetries in two directions. The nontrivial topology for all these systems is further confirmed from a calculation of the topological invariant. Associated transport properties and second order effects have also been studied for these systems. The non-conservation terms of the energy momentum tensor could come from an external effective symmetric tensor matter field or a gravitational field which could be generated by a specific non-inertial reference frame transformation from the original inertial reference frame. Finally we introduce a possible holographic realization of one of these systems. We propose a new method to calculate the holographic Ward identities for the energy momentum tensor without calculating out all components of the Green functions and match the Ward identities of both sides.

53 pages, 19 figures; v3: matches the published version