If two objects A and B of different size are both at a temperature of 500 K and are connected by a rod as in the image below. What will be the nature of transmission of heat?

Heat will flow from the object at a higher temperature (500 K) to the object at a lower temperature along the rod until both objects reach thermal equilibrium. The rate at which heat is transferred will depend on the thermal conductivity and surface area of the rod connecting the two objects.