I am trying to find least cost paths from the southern USA to the Great Lakes region (for example). My resistance layers are currently in an equal area projection but I'm wondering if I should convert to an equidistant projection? My understanding is that part of what factors into a least-cost path analysis is distance (all else being equal, the shortest route from A to B is the least-costly) and so to me it would make sense to have the raster layers in an equidistant projection...but maybe I'm missing something? If equidistance does make sense, what is the most appropriate for all of North America?
The problems arise because the CostDistance calculations use the (Euclidean) distance in a map as a surrogate for the true distances experienced on the globe's surface. This surrogate will be distorted in two ways:
The relationship between map distance and globe distance will vary according to location on the map.
At any given point, the map-globe distance relationship (aka the scale) may vary depending on the bearing between two points.
It is impossible to avoid the first problem (although well-chosen projections for maps covering smallish regions, such as the size of a few states, will have so little distortion that one usually doesn't worry). However, it is possible to avoid the second one altogether. There is a class of projections where the scale does not depend on the bearing: these are known as conformal projections.
You can, therefore, achieve high accuracy by choosing a conformal projection and adjusting the impedances to compensate for the scale variation across the map. This is done in exactly the same way one would adjust slope calculations in a DEM. I describe how distances can be adjusted at https://gis.stackexchange.com/a/58114 (for the purpose of interpolation, which has much in common with CostDistance calculations). I detail the workflow and give appropriate formulas at https://gis.stackexchange.com/a/40464. That post suggests choosing a conformal projection whose scale distortion is relatively simple to calculate. That leads to simple projections such as the Mercator or Stereographic. Even though these may introduce relatively large changes in scale across a large area, that is more than compensated by the ability to adjust the impedance.
To be specific, areas of the map that make distances look larger than suggested by the nominal scale by some factor f must have their impedances divided by f to decrease them. Areas of the map that make distances look smaller will have their impedances increased. If, for instance, you select a Mercator projection, the impedances must be decreased the further North you go because that projection makes distances seem comparatively greater.
There is no need to work really hard at achieving high accuracy with CostDistance calculations, because they all introduce some inherent inaccuracy from their discretization of bearings into just eight directions (left-right, up-down, and diagonally). The important thing is to avoid any bias that is sufficiently large and sufficiently local (that is, focused on just one part of the study area) that is might change the optimal solutions in substantial ways. That is why it is not necessary to consider non-spherical models of the earth (with their attendant complications).
Equidistant projections are not equidistant from all locations to all locations. As Vince states, if all your origins are (close to) centered on one location you can use the "equidistant" projection below and center it on your origin. If not then stick with the equal Albers Area USA. A would do test run in both.
A tutorial here shows how to center it on Gainseville as an example