ありがとう/ 有難う (arigatou).

Yesterday a friend was asking me whether arigatou is written with kanji or kana. I said I'd never seen it in kanji but assumed that at one point kanji were associated with it, whether for meaning or just for pronunciation.

Turns out the old kanji are a good match for meaning: 有難う,　有 being to be or exist and 難 the same kanji as for 難しい, difficult. 有り難い / 有難い is apparently an adjective that involves gratitude, but it's a new one to me. Can anyone explain how 有難い becomes 有難う? Can other -い adjectives use -おう? Eg, does 違おう mean anything? 暖こう? 寒おう？ 若う？食べとう？ Or is there no pattern here?

ADDENDUM: Google does show examples of 暖こうございます and 食べとうおざる. If there is an -ou form, maybe it refers to a state that the speaker is or was in.

ADDENDUM 2: According to this,

The "ou" ending comes from the conjunction of the adjectival arigataku with the polite verb gozaimasu.... Other common examples of this type of conjugation include omedetou gozaimasu (congratulations) from medetaku and ohayou gozaimasu (good morning) from hayaku. The word arigatai existed in Japanese long before the Japanese ever encountered Portuguese. It can be found in some of the earliest Japanese literature such as the manyoushuu....

Now, if I can just figure out why a -く form would precede である....