  A theoretical understanding of the enigmatic linear-in-temperature ($T$)
resistivity, ubiquitous in strongly correlated metallic systems, has been a
long sought-after goal. Furthermore, the slope of this robust $T$-linear
resistivity is also observed to stay constant through crossovers between
different temperature regimes: a phenomenon we dub "slope invariance".
Recently, several solvable models with $T$-linear resistivity have been
proposed, putting us in an opportune moment to compare their inner workings in
various explicit calculations. We consider two strongly correlated models with
local self-energies that demonstrate $T$-linearity: a lattice of coupled
Sachdev-Ye-Kitaev (SYK) models and the Hubbard model in single-site dynamical
mean-field theory (DMFT). We find that the two models achieve $T$-linearity
through distinct mechanisms at intermediate temperatures. However, we also find
that these mechanisms converge to an identical form at high temperatures.
Surprisingly, both models exhibit "slope invariance" across the two temperature
regimes. We thus not only reveal some of the diversity in the theoretical inner
workings that can lead to $T$-linear resistivity, but we also establish that
different mechanisms can result in "slope invarance".
