  We show that thermal states of local Hamiltonians are separable above a
constant temperature. Specifically, for a local Hamiltonian $H$ on a graph with
degree $\mathfrak{d}$, its Gibbs state at inverse temperature $\beta$, denoted
by $\rho =e^{-\beta H}/ \textrm{tr}(e^{-\beta H})$, is a classical distribution
over product states for all $\beta < 1/(c\mathfrak{d})$, where $c$ is a
constant. This sudden death of thermal entanglement upends conventional wisdom
about the presence of short-range quantum correlations in Gibbs states.
  Moreover, we show that we can efficiently sample from the distribution over
product states. In particular, for any $\beta < 1/( c \mathfrak{d}^3)$, we can
prepare a state $\epsilon$-close to $\rho$ in trace distance with a depth-one
quantum circuit and $\textrm{poly}(n) \log(1/\epsilon)$ classical overhead. A
priori the task of preparing a Gibbs state is a natural candidate for achieving
super-polynomial quantum speedups, but our results rule out this possibility
above a fixed constant temperature.
