In Constraint Programming, are there any models that take into account the number of variable changes? Consider a CSP model where changing the value of a particular variable is expensive. Is there any work where the objective function also considers the number of changes in the value of the variable during the search process?

An example: The expensive-to-change variable may be in the control of some other agent and there is some overhead of involving that agent to change the variable. Another example: The variable participates in one of the constraints, and the satisfaction of this constraint involves calling an expensive function (such as, a simulator), e.g. $z = f(x, y)$ is the constraint, and $f$ is an expensive-to-compute function. Therefore, $x$ and $y$ are expensive-to-change variables.
It sounds like you want a cost-sensitive (cost-aware, budgeted) optimization technique. Minimizing two values (e.g. the solution of your objective and the cost of operations on $x$ and $y$) is a multicriteria optimization problem, and those tend to be very hard to solve. A common approach is to specify a budget for the maximum allowable costs and then minimize the objective function with respect to $costs(x,y) \le Budget$. This formulation tends to fit nicely into existing frameworks as an additional constraint. Of course, specifying the cost function and the allowable budget in such a way that you get meaningful solutions can be difficult - this will depend on the specific problem you are trying to solve.