Given the living conditions of archaea, what mechanisms do they have to regulate homeostasis?

Although there are mesophile archaea (live in mild conditions), most of the archaea live in extreme conditions (extremophiles). These extreme conditions may be related to high salt concentrations (halophiles), high temperatures (thermophiles), very alkaline conditions (alkaliphiles), or very acidic conditions (acidophiles).
In order to regulate homeostasis:
Halophiles uses aerobic respiration in waters that can go up to 25%NaCl concentration.
Thermophiles have heat shock proteins whose purpose is to refold denatured proteins to their functional conformation. These heat shock proteins also restore damaged proteins. There are several other mechanisms aiming to protect DNA. For example, some archaea have a protein that stabilises the DNA double-helix conformity raising its DNA melting point by 104ºF. Also, the cell membrane structure is heat-resistant - instead of simple phospholipids, their membranes are made of di-bi-phytanyl diether lipids.
Acidophiles can use carbon dioxide for as a carbon source. They can also, for example, in environments with oxygen process organic molecules by using TCA cycle reversely in order to convert sulfur and oxygen into sulfuric acid.

Amutation is basically the change in the gene, caused by alteration of single based units in the dna, or the genes or chromosomes were rearranged in the dna