  We introduce a new deformation of multiple zeta value (MZV). It has one
parameter $\omega$ satisfying $0<\omega<2$ and recovers MZV in the limit as
$\omega \to +0$. It is defined in the same algebraic framework as a
$q$-analogue of multiple zeta value ($q$MZV) by using a multiple integral. We
prove that our deformed multiple zeta value satisfies the double shuffle
relations which are satisfied by $q$MZVs. We also prove the extended double
Ohno relations, which are proved for ($q$)MZVs by Hirose, Sato and Seki, by
using a multiple integral whose integrand contains the hyperbolic gamma
function due to Ruijsenaars.
