Black holes in Einstein-Gauss-Bonnet gravity with a background of modified Chaplygin gas

Supposing the existence of modified Chaplygin gas with the equation of state $p=A\rho-B/\rho^\beta$ as a cosmic background, we obtain a static spherically-symmetric solution to the Einstein-Gauss-Bonnet gravitational equations in 5D spacetime. The spacetime structure of the obtained black hole solution could be asymptotically anti-de Sitter or de Sitter, according to the specific values of modified Chaplygin gas parameters versus the cosmological constant. We analyze the parametric regions for both kinds of solutions. For asymptotically anti-de Sitter black hole, there exists the so-called small/large black hole phase transition, we obtain critical values of pressure, volume, and temperature and investigate the effects of both the Gauss-Bonnet gravity and the modified Chaplygin gas on these values. For asymptotically de Sitter black hole, no $P-r_h$ criticality and phase transition appear, and we show that the thermodynamic systems related to various horizons of asymptotically de Sitter black hole are in fact entangled.