The relativistic quantum interference effects in the spacetime of slowly rotating object in the Hořava-Lifshitz gravity as the Sagnac effect and phase shift of interfering particle in neutron interferometer are derived . We consider the extension of Kehagias-Sfetsos ( KS ) solution [ 44 ] in the Hořava-Lifshitz gravity for the slowly rotating gravitating object . Using the covariant Klein-Gordon equation in the nonrelativistic approximation , it is shown that the phase shift in the interference of particles includes the gravitational potential term with the KS parameter \omega . It is found that in the case of the Sagnac effect , the influence of the KS parameter \omega is becoming important due to the fact that the angular velocity of the locally non rotating observer is increased in Hořava gravity . From the results of the recent experiments [ 46 ] we have obtained lower limit for the coupling KS constant as \omega \simeq 1.25 \cdot 10 ^ { -25 } cm ^ { 2 } . Finally , as an example , we apply the obtained results to the calculation of the UCN ( ultra-cold neutrons ) energy level modification in the gravitational field of slowly rotating gravitating object in the Hořava-Lifshitz gravity .