Recently published discoveries of acoustic- and optical-mode inversion in the phonon spectrum of certain metals became the first realistic example of noninteracting topological bosonic excitations in existing materials. However, the observable physical and technological use of such topological phonon phases remained unclear. In this paper, we provide strong theoretical and numerical evidence that for a class of metallic compounds (known as triple-point topological metals), the points in the phonon spectrum, at which three (two optical and one acoustic) phonon modes (bands) cross, represent a well-defined topological material phase, in which the hosting metals have very strong thermoelectric response. The triple-point bosonic collective excitations appearing due to these topological phonon band-crossing points significantly suppress the lattice thermal conductivity, making such metals phonon glasslike. At the same time, the topological triple-point and Weyl fermionic quasiparticle excitations present in these metals yield good electrical transport (electron crystal) and cause a local enhancement in the electronic density of states near the Fermi level, which considerably improves the thermopower. This combination of phonon glass and electron crystal is the key for high thermoelectric performance in metals. We call these materials topological thermoelectric metals and propose several compounds for this phase (TaSb and TaBi). We hope that this work will lead researchers in physics and materials science to the detailed study of topological phonon phases in electronic materials, and the possibility of these phases to introduce more efficient use of thermoelectric materials in many everyday technological applications.