  We derive model-independent astrophysical constraints on leptophilic dark
matter (DM), considering its thermal production in a supernova core and taking
into account core temperature fluctuations within the framework of $q$-deformed
Tsallis statistics. In an effective field theory approach, where the DM
fermions interact with the Standard Model via dimension-six operators of either
scalar-pseudoscalar, vector-axial vector, or tensor-axial tensor type, we
obtain bounds on the effective cut-off scale $\Lambda$ from supernova cooling
and free-streaming of DM from supernova core, and from thermal relic density
considerations, depending on the DM mass and the $q$-deformation parameter.
Using Raffelt's criterion on the energy loss rate from SN1987A, we obtain a
lower bound on $\Lda \gtrsim 3$ (12) TeV corresponding to $q = 1.0~(1.1)$ and
an average supernova core temperature of $T_{\rm SN}=30$ MeV. From the optical
depth criterion on the free-streaming of DM fermions from the outer 10\% of the
SN1987A core, the cooling bound is restricted to $\Lda \gtrsim 1$ TeV. Both
cooling and free-streaming bounds are insensitive to the DM mass $m_\chi$ for
$m_\chi\lesssim T_{\rm SN}$, whereas for $m_\chi\gg T_{\rm SN}$, the bounds
weaken significantly due to the Boltzmann-suppression of the DM number density.
We also calculate the thermal relic density of the DM particles in this setup
and find that it imposes an upper bound on $\Lambda^4/m_\chi^2$, which together
with the cooling/free-streaming bound significantly constrains light
leptophilic DM.
