Motivated by recent work on rotating black hole shadow (Chang and Zhu in Phys Rev D 101:084029, 2020), we investigate the shadow behaviours of rotating Hayward–de Sitter black hole for static observers at a finite distance in terms of astronomical observables. This paper uses the newly introduced distortion parameter (Chang and Zhu in Phys Rev D 102:044012, 2020) to describe the shadow’s shape quantitatively. We show that the spin parameter would distort shadows and the magnetic monopole charge would increase the degree of deformation. The distortion will increase as the distance between the observer and the black hole increases, and distortion reduces as the cosmological constant increases. Besides, the increase of the spin parameter, magnetic monopole charge and cosmological constant will cause the shadows shrunken.

In this article, we have investigated the equations of motion of the photons coupled to Weyl tensor by the geometric optics approximation and the corresponding shadow in a Brane-World black hole spacetime. It is shown that there exists a double shadow for a black hole since the coupling photons with different polarization directions propagate along different paths in the spacetime. Furthermore, we discuss the effects of the metric parameter α related to the cosmological constant, X-cold dark matter parameter β and the coupling parameter λ on the umbra (the overlap region of the double shadow) and the penumbra. We also obtain the finite-distance corrections to the deflection angle of light in the Brane-World black hole spacetime as the photons coupled to Weyl tensor by using a recent geometric method.

The thermodynamics of the Universe in Eddington-Born-Infeld (EBI) theory is restudied by utilizing the holographic-style gravitational equations that dominate the dynamics of the cosmical apparent horizon Υ A and the evolution of Universe. We started from applying the Bigravity method to rewrite the EBI action of the Palatini approach into the Bigravity-type action with an extra metric q µν . With the help of the holographic-style dynamical equations, we discussed the properties of the cosmical apparent horizon Υ A including timelike, spacelike and null characters, which depend on the value of the parameter of state w m in EBI Universe. We also obtained the unified first law for the gravitational thermodynamics and the total energy differential for the open system enveloped by Υ A in EBI Universe. Finally, we used the Gibbs equation in the positive-heat-out sign convention to derive the generalized second laws of the nondecreasing entropy S (A) tot enclosed by Υ A in EBI universe.