Set up a style for the object that has the display property turned off for each element and specify that style for the object. Several null display object styles are included in the templates provided by Autodesk if you are using one of those or want to copy or use one for reference in creating your own.
Layers and "freeze-thaw" "on-off" states are much less useful or predictable with Civil 3D - especially when picking an entity to alter it's layer state. It works just often enough to keep you trying but can have very unintended consequences not readily apparent. Layer 0 is especially vulnerable and often the only remedy will unravel the display changes you made. I must emphasize that the picking of an entity to freeze or otherwise alter with regular AutoCAD functions is best avoided.
Each Civil 3D object can be assigned a variety of styles that alter the visibility, color line type, line weight, object style, etc. of every entity available to assign to that object. If an object is exported as a data reference, the drawing(s) into which it is imported can assign any available style, regardless of how it is displayed (or not) in the parent drawing. The same surface within a set of drawings could at the same time be seen as 2' contours, 5' contours, flow direction arrows, color coded elevation zones, a simple boundary to show its extents and nothing at all. The same is basically true for all objects, though with perhaps fewer options.
