For a class of robust convex multiobjective optimization problem with uncertain data，a robust counterpart under general uncertain sets is given. The scalar quantization method is used to transform the robust multiobjective counterpart into a robust single object convex optimization problem，to establish the connection between the two solutions. And we obtain a multiplicative description of a scalarized robust solution and the generalized constant differential property and constant Lagrangian property of the scalarization problem on its robust solution set. Finally，through the preceding properties，the proper efficient solution set of a robust convex multiobjective optimization problem is characterized and proved.