  In this article, we derive rogue wave (RW) solutions of a fifth-order
nonlinear Schr\"odinger equation over a double-periodic wave background.
Choosing the elliptic functions (combinations of $cn$, $dn$ and $sn$) as seed
solutions in the first iteration of Darboux transformation and utilizing the
nonlinearization of Lax pair procedure, we create the double-periodic wave
background for the fifth-order nonlinear Schr\"odinger equation. By introducing
the second linearly independent solution, we generate the RW solutions on the
created background for three different eigenvalues. We demonstrate the
differences that occur in the appearance of RWs due to the lower-order and
higher-order dispersions terms. We examine the derived solution in detail for
certain system and elliptic modulus parameters values and highlight some
interesting features that we obtain from our studies. We also calculate the
growth rate for instability of double-periodic solutions under different values
of elliptic modulus parameter.
