  Charge density wave (CDW) correlations are prevalent in all copper-oxide
superconductors. While CDWs in conventional metals are driven by coupling
between lattice vibrations and electrons, the role of the electron-phonon
coupling (EPC) in cuprate CDWs is strongly debated. Using Cu $L_3$ edge
resonant inelastic x-ray scattering (RIXS), we study the CDW and Cu-O
bond-stretching phonons in the stripe-ordered cuprate
La$_{1.8-x}$Eu$_{0.2}$Sr$_{x}$CuO$_{4+\delta}$. We investigate the interplay
between charge order and EPC as a function of doping and temperature, and find
that the EPC is enhanced in a narrow momentum region around the CDW wave
vector. By detuning the incident photon energy from the absorption resonance,
we extract an EPC matrix element at the CDW wave vector of $M\simeq$ 0.36 eV,
which decreases to $M\simeq$ 0.30 eV at high temperature in the absence of the
CDW. Our results suggest a feedback mechanism in which the CDW enhances the EPC
which, in turn, further stabilizes the CDW.
