Calculate cost of predefined path

Using GRASS GIS, you could convert the path to a raster, then extract the slope values from all slope pixels along the path raster, and sum them up. So:

# Assuming a line vector named "path" and cost raster named "slope"
v.to.rast input=path output=path type=line use=val value=1
r.mapcalc "path_cost = if (path=1, slope, null())"
# The sum of the pixels will be the total cost based on slope of all pixels "under" the path
r.univar path_cost # Check value of "sum"