  The Sachdev-Ye-Kitaev (SYK) model describes electrons with random and
all-to-all interactions, and realizes a many-body state without quasiparticle
excitations, and a non-vanishing extensive entropy $S_0$ in the zero
temperature limit. Its low energy theory coincides the low energy theory of the
near-extremal charged black holes with Bekenstein-Hawking entropy $S_0$.
Several mesoscopic experimental configurations realizing SYK quantum dynamics
over a significant intermediate temperature scale have been proposed. We
investigate quantum thermoelectric transport in such configurations, and
describe the low temperature crossovers out of SYK criticality into regimes
with either Fermi liquid behavior, a Coulomb blockade, or criticality
associated with Schwarzian quantum gravity fluctuations. Our results show that
thermopower measurements can serve as a direct probe for $S_0$.
