  The current fluctuations due to a temperature bias, i.e. the delta-$T$ noise,
allow one to access properties of strongly interacting systems which cannot be
addressed by the usual voltage-induced noise. In this work, we study the full
delta-$T$ noise between two different fractional quantum Hall edge states, with
filling factors $(\nu_L,\nu_R)$ in the Laughlin sequence, coupled through a
quantum point contact and connected to two reservoirs at different
temperatures. We are able to solve exactly the problem for all couplings and
for any set of temperatures in the specific case of an hybrid junction
$(1/3,1)$. Moreover, we derive a universal analytical expression which connects
the delta-$T$ noise to the equilibrium one valid for all generic pairs
$(\nu_L,\nu_R)$ up to the first order in the temperature mismatch. We expect
that the linear term can be accessible in nowadays experimental set-ups. We
describe the two opposite coupling regimes focusing on the strong one which
correspond to a non-trivial situation. Our analysis on delta-$T$ noise allows
us to better understand the transport properties of strongly interacting
systems and to move toward more involved investigation concerning the
statistics and scaling dimension of their emergent excitations.
