This paper presents a novel approach to design RF tunable bandstop filters with constant absolute bandwidth. The filters are based on a doublet configuration which is attractive for high-order filter designs. In the doublet, two varactor-tuned resonators, arranged in a mirror-symmetrical manner, are coupled to a main transmission line. A novel coupling scheme is utilized between the resonators and the main line. Theoretical analysis indicates that, as the resonant frequency varies, the coupling coefficient within the fixed coupling region can be inherently changed to desirable values to meet the requirement of constant absolute bandwidth. Using the coupling scheme, constant absolute bandwidth can be obtained without extra circuits to control the coupling strengths. For demonstration, first-order and high-order filters are designed. Comparisons of experimental and simulated results are presented to verify the theoretical predications.