We test the anisotropy in the Finslerian cosmological model with the X-ray and ultraviolet (UV) fluxes of 808 quasars. The dipole amplitude is $A_D=0.302_{ -0.124}^{ +0.185}$ and the dipole direction points towards $(l, b) = ( 288.92_{~ -28.80^{circ}}^{^{circ}+23.74^{circ}}, 6.10_{~ -16.40^{circ}}^{^{circ} +16.55^{circ}} )$. We find that the dipole direction from the X-ray and UV fluxes of quasars is very close to the dipole direction given by the Joint Light-curve Analysis (JLA) compilation in the Finslerian cosmological model and the angular difference between the two dipole directions is only $10.44^{circ}$. We also find the angular difference between the dipole direction from the 808 quasars in the Finslerian cosmological model and ones from the supernovae of type Ia (SNe Ia) samples in the dipole-modulated $Lambda$CDM model is around $30^{circ}$. Six gravitationally lensed quasars are considered to investigate the Hubble constant $H_0$ in the Finslerian cosmological model. We get a slightly smaller $H_0$ than the result given by the six gravitationally lensed quasars. Finally, we forecast the future constraints on the dipole parameters with the X-ray and UV fluxes of quasars. As the number of simulations increases, the precisions of the parameters related to anisotropy in the Finslerian cosmological model improve significantly. The X-ray and UV fluxes of quasars have a promising future as a probe of anisotropy in Finsler spacetime.
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