  A novel H3N3-2$_\sigma$ interpolation approximation for the Caputo fractional
derivative of order $\alpha\in(1,2)$ is derived in this paper, which improves
the popular L2C formula with (3-$\alpha$)-order accuracy. By an interpolation
technique, the second-order accuracy of the truncation error is skillfully
estimated. Based on this formula, a finite difference scheme with second-order
accuracy both in time and in space is constructed for the initial-boundary
value problem of the time fractional hyperbolic equation. It is well known that
the coefficients' properties of discrete fractional derivatives are fundamental
to the numerical stability of time fractional differential models. We prove the
related properties of the coefficients of the H3N3-2$_\sigma$ approximate
formula. With these properties, the numerical stability and convergence of the
difference scheme are derived immediately by the energy method in the sense of
$H^1$-norm. Considering the weak regularity of the solution to the problem at
the starting time, a finite difference scheme on the graded meshes based on
H3N3-2$_\sigma$ formula is also presented. The numerical simulations are
performed to show the effectiveness of the derived finite difference schemes,
in which the fast algorithms are employed to speed up the numerical
computation.
