We use gauge-gravity duality to study the temperature dependence of the zero sound mode and the fundamental matter diffusion mode in the strongly coupled $\mathcal{N}=4$ $SU(N_c)$ supersymmetric Yang-Mills theory with $N_f$ $\mathcal{N}=2$ hypermultiplets in the $N_c\gg 1$, $N_c\gg N_f$ limit, which is holographically realized via the D3/D7 brane system. In the high density limit $\mu \gg T$, three regimes can be identified in the behavior of these modes, analogous to the collisionless quantum, collisionless thermal, and hydrodynamic regimes of a Landau Fermi liquid. The transitions between the three regimes are characterized by the parameters $T/\mu$ and $(T/\mu {)}^2$, respectively, and in each of these regimes the modes have a distinctively different temperature and momentum dependence. The collisionless-hydrodynamic transition occurs when the zero sound poles of the density-density correlator in the complex frequency plane collide on the imaginary axis to produce a hydrodynamic diffusion pole. We observe that the properties characteristic of a Landau Fermi-liquid zero sound mode are present in the D3/D7 system despite the atypical $T^6/{\mu }^3$ temperature scaling of the specific heat and an apparent lack of a directly identifiable Fermi surface.

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