Yes, the friction coefficient stays the same, but not the rolling resistance
Furthermore, the friction coefficient have nothing to do with the effort you have to give. The friction only reacts to how much force you put at the tire contact patch. The friction coefficient represents the maximum
force you CAN put on the road.
This is not the case with rolling resistance coefficient which represents the actual
force you have to fight to keep moving forward.
Find more about the subject on this link
(It uses car to explain the theory, but everything applies to bicycle as well). Check the theory tab at the bottom of the page.