  The cold dark matter fermions with mass MeV scale, pair produced inside the
supernova SN1987A core, can freely stream away from the supernovae and hence
contributes to its energy loss rate. Similar type of DM fermions(having similar
kind of coupling to the standard model photon), produced from some other
sources earlier, could have contributed to the relic density of the Universe.
Working in a theory with an effective dark matter-photon coupling (inversely
proportional to the scale $\Lambda$) in the formalism of Tsallis statistics, we
find the dark matter contribution to the relic density and obtain a upper bound
on $\Lambda$ using the experimental bound on the relic density for cold
non-baryonic matter i.e. $\Omega h^2 = 0.1186 \pm 0.0020 $. The upper bound
obtained from the relic density is shown with the lower bound obtained from the
Raffelt's criterion on the emissibity rate of the supernovae SN1987A energy
loss $\dot{\varepsilon}(e^+ e^- \to \chi \overline{\chi}) \le
10^{19}~\rm{erg~g^{-1}s^{-1}}$ and the optical depth criteria on the free
streaming of the dark matter fermion (produced inside the supernovae core). As
the deformation parameter $q$ changes from $1.0$ (undeformed scenario) to
$1.1$(deformed scenario), the relic density bound on $\Lambda$ is found to vary
from $ \sim 4.9 \times 10^7 $ TeV to $1.6 \times 10^8$ TeV for a fermion dark
matter($\chi$) of mass $m_\chi = 30~\rm{MeV}$, which is almost $10$ times more
than the lower bound obtained from the SN1987A energy loss rate and the optical
depth criteria.
  \noindent {{\bf Keywords}: Dark matter, Relic density, Supernova cooling,
Tsallis statistics, free-streaming, } }
