Study of <math><mi>C</mi><mi>P</mi></math> violation in <math><msup><mi>D</mi><mo>±</mo></msup><mo>→</mo><msup><mi>K</mi><mo>*</mo></msup><mo>(</mo><mn>892</mn><msup><mo>)</mo><mn>0</mn></msup><msup><mi>π</mi><mo>±</mo></msup><mo>+</mo><msup><mover><mi>K</mi><mo>¯</mo></mover><mo>*</mo></msup><mo>(</mo><mn>892</mn><msup><mo>)</mo><mn>0</mn></msup><msup><mi>π</mi><mo>±</mo></msup><mo>→</mo><msubsup><mi>K</mi><mrow><mi>S</mi><mo>,</mo><mi>L</mi></mrow><mn>0</mn></msubsup><msup><mi>π</mi><mn>0</mn></msup><msup><mi>π</mi><mo>±</mo></msup></math> decays

Within the Standard Model, we investigate the CP violations and the KS0−KL0 asymmetries in D&#177;→K*(892)0π&#177;+K&#175;*(892)0π&#177;→KS,L0π0π&#177; decays basing on the factorization-assisted topological-amplitude (FAT) approach and the topological amplitude (TA) approach of Cheng and Chiang [Phys. Rev. D 104, 073003 (2021).]. We find that the CP violations in these decays ACPKS,L0 can exceed the order of 10−3 in the two approaches and consist of three parts: the indirect CP violations in K0−K&#175;0 mixing ACP,KS,L0mix, the direct CP violations in charm decays ACP,KS,L0dir, and the new CP violation effects ACP,KS,L0int, which are induced from the interference between two tree (Cabibbo-favored and doubly Cabibbo-suppressed) amplitudes with the neutral kaon mixing. The indirect CP violations in K0−K&#175;0 mixing play a dominant role in ACPKS,L0; the new CP violation effects have a non-negligible contribution to ACPKS,L0. We estimate the numerical results of the KS0−KL0 asymmetries RKS−KLD&#177; and find that there exists a large difference between the numerical results in the FAT approach and that of the TA approach. We present the numbers of D&#177; events-times-efficiency needed to observe the CP violations and the KS0−KL0 asymmetries at the level of 3 standard deviations (3σ). We also find that if one adopts the values of the decay time parameters t0=3.0τS and t1=10.0τS, the new CP violation effect ACP,KS0int would dominate the CP violation in D&#177;→K*(892)0π&#177;+K&#175;*(892)0π&#177;→KS0π0π&#177; decays and could be observed with 6.7&#215;106 and 6.5&#215;106 D&#177; events-times-efficiency in the FAT approach and the TA approach, respectively. Our results could be tested by the LHCb (Large Hadron Collider beauty), Belle II, and BESIII experiments.