We report the first result of a direct search for a cosmic axion background (CaB)-a relativistic background of axions that is not dark matter-performed with the axion haloscope, the Axion Dark Matter eXperiment (ADMX). Conventional haloscope analyses search for a signal with a narrow bandwidth, as predicted for dark matter, whereas the CaB will be broad. We introduce a novel analysis strategy, which searches for a CaB induced daily modulation in the power measured by the haloscope. Using this, we repurpose data collected to search for dark matter to set a limit on the axion photon coupling of a CaB originating from dark matter cascade decay via a mediator in the 800-995 MHz frequency range. We find that the present sensitivity is limited by fluctuations in the cavity readout as the instrument scans across dark matter masses. Nevertheless, we suggest that these challenges can be surmounted using superconducting qubits as single photon counters, and allow ADMX to operate as a telescope searching for axions emerging from the decay of dark matter. The daily modulation analysis technique we introduce can be deployed for various broadband rf signals, such as other forms of a CaB or even high-frequency gravitational waves.