The interplay of strong atomic spin-orbit coupling with Jahn-Teller (JT) lattice distortions is an important theme in quantum materials hosting heavy atoms. A prototypical example of such a system is a degenerate $j=3/2$ multiplet coupled to local phonon modes. Here, we study the multipolar ground states of this system as realized in Mott insulators, explore its thermal phase diagram via an $SU\left(4\right)$ spin Monte Carlo approach coupled to JT phonons, and study the resulting temperature dependent spectral distribution of the $j=3/2$ multiplet. Our work sheds light on coexisting distinct multipolar orders engendered by phonon coupling, role of thermal JT fluctuations, and the entropy of the coupled multipole-phonon system. We discuss broad implications for double perovskites ${\mathrm{Ba}}_2{\mathrm{MgReO}}_6, {\mathrm{Ba}}_2{\mathrm{NaOsO}}_6$ and lacunar spinels such as ${\mathrm{GaTa}}_4{\mathrm{Se}}_8$ and ${\mathrm{GaNb}}_4{\mathrm{Se}}_8$.

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• Frustrated magnetism
• Jahn-Teller effect
• Magnetism
• Multipole order
• Spin-phonon coupling
• Monte Carlo methods