In a number of graph search-based planning problems, the value of the cost function that is being minimized also affects the set of possible actions at some or all the states in the graph. For example, in path planning for a robot with a limited battery power, a common cost function is energy consumption, whereas the level of remaining energy affects the navigational capabilities of the robot. Similarly, in path planning for a robot navigating dynamic environments, a total traversal time is a common cost function whereas the timestep affects whether a particular transition is valid. In such planning problems, the cost function typically becomes one of the state variables thereby increasing the dimensionality of the planning problem, and consequently the size of the graph that represents the problem. In this paper, we show how to avoid this increase in the dimensionality for the planning problems whenever the availability of the actions is monotonically non-increasing with the increase in the cost function. We present three variants of A* search for dealing with such planning problems: a provably optimal version, a suboptimal version that scales to larger problems while maintaining a bound on suboptimality, and finally a version that relaxes our assumption on the relationship between the cost function and action space. Our experimental analysis on several domains shows that the presented algorithms achieve up to several orders of magnitude speed up over the alternative approaches to planning.