  We show that the bipartite logarithmic entanglement negativity (EN) of
quantum spin models obeys an area law at all nonzero temperatures. We develop
numerical linked cluster (NLC) expansions for the `area-law' logarithmic
entanglement negativity as a function of temperature and other parameters. For
one-dimensional models the results of NLC are compared with exact
diagonalization on finite systems and are found to agree very well. The NLC
results are also obtained for two dimensional XXZ and transverse-field Ising
models. In all cases, we find a sudden onset (or sudden death) of negativity at
a finite temperature above which the negativity is zero. We use perturbation
theory to develop a physical picture for this sudden onset (or sudden death).
The onset of EN or its magnitude are insensitive to classical
finite-temperature phase transitions, supporting the argument for absence of
any role of quantum mechanics at such transitions. On approach to a quantum
critical point at $T=0$, negativity shows critical scaling in size and
temperature.
