KOI-13 is the first known transiting system exhibiting light curve distortions due to gravity darkening of the rapidly rotating host star .
In this paper we analyse publicly available Kepler Q2–Q3 short-cadence observations , revealing a continuous light variation with a period of P _ { rot } = 25.43 \pm 0.05 hour and a half-amplitude of 21 ppm , which is linked to stellar rotation .
This period is in exact 5:3 resonance with the orbit of KOI-13.01 , which is the first detection of a spin-orbit resonance in a host of a substellar companion .
The stellar rotation leads to stellar oblateness , which is expected to cause secular variations in the orbital elements .
We indeed detect the gradual increment of the transit duration with a rate of ( 1.14 \pm 0.30 ) \times 10 ^ { -6 } day/cycle .
The confidence of this trend is 3.85- \sigma , the two-sided false alarm probability is 0.012 % .
We suggest that the reason for this variation is the expected change of the impact parameter , with a rate of { d } b / { d } t = -0.016 \pm 0.004 / { yr } .
Assuming b \approx 0.25 , KOI-13.01 may become a non-transiting object in 75 - 100 years .
The observed rate is compatible with the expected secular perturbations due to the stellar oblateness yielded by the fast rotation .