I am using the cost distance tool to map out helicopter response times with constraints (mountain passes, geographic boundaries, etc). I have a uniform cost raster where the value of each pixel is equal to the amount of time it takes the helicopter to overfly the pixel. The Cost Distance Tool adds up all these bits of time to produce a map of response times, but where I would expect the map to be circular in shape it instead is an 8-sided polygon. My question...

Is there a way to make Cost Distance Output maps circular, or are polygon outputs the result of the underlying algorithm?

See this discussion for a nice explanation of exactly your problem (in the end, use Euclidean Distance, as mentioned above):

Circular vs. Octagonal Cost Distance