Edit: i up 2 AWs and the cost does not change, i do not understand how the cost is calculated !
Also, if you're only looking at cost changes you may not notice differences due to rounding. If game calculates 411K gold or 413K gold per negotiation, it will round both to 410K so you won't see a difference. To see finer details, you need to look at Spire squad sizes, these are only rounded to integers.costs are fixed at the moment of spire start. You'll see the difference next week.
It depends not only on AW levels, it includes tech tree progress/provinces finished/etc
Based on my observations so far, base Spire squad sizes increase by about 6.5 and 2.9 (yours / enemy's) per extra AW level. The rest of the cost increases can be recalculated from that. But I'll need more data to figure out if these are absolute numbers, or specific to the rest of my setup (e.g. provinces etc). I'll post a separate thread asking for more data.