Cloud storage is vulnerable to advanced persistent threats (APTs), in which an attacker launches stealthy, continuous, well-funded and targeted attacks on storage devices. In this paper, cumulative prospect theory (CPT) is applied to study the interactions between a defender of cloud storage and an APT attacker when each of them makes subjective decisions to choose the scan interval and attack interval, respectively. Both the probability weighting effect and the framing effect are applied to model the deviation of subjective decisions of end-users from the objective decisions governed by expected utility theory, under uncertain attack durations. Cumulative decision weights are used to describe the probability weighting effect and the value distortion functions are used to represent the framing effect of subjective APT attackers and defenders in the CPT-based APT defense game, rather than discrete decision weights, as in earlier prospect theoretic study of APT defense. The Nash equilibria of the CPT-based APT defense game are derived, showing that a subjective attacker becomes risk-seeking if the frame of reference for evaluating the utility is large, and becomes risk-averse if the frame of reference for evaluating the utility is small.