Store previous temperature for non-ideal EoS

Out of curiosity - is it possible for the T(e) function to use temperature values from the previous timestep as initial guess (if those are stored as auxiliary variables)? Not necessarily now, but wondering if that's something that could be added in the future. Haven't really been keeping up with the changes introduced in this PR, although I'm trying to follow :)

_Originally posted by @knstmrd in https://github.com/trixi-framework/Trixi.jl/pull/3079#discussion_r3476899047_

No file, test, or entry point is named. Start by locating the T(e) function, the non-ideal EoS path, and auxiliary-variable handling; then determine whether previous-timestep temperatures can be used as initial guesses and define tests for the resulting behavior.

Written by the indexing model from the issue text.