  We theoretically investigate the critical current of a thermally-biased SIS
Josephson junction formed by electrodes made by different BCS superconductors.
The response of the device is analyzed as a function of the asymmetry
parameter, $r=T_{c_1} /T_{c_2}$. We highlight the appearance of jumps in the
critical current of an asymmetric junction, namely, when $r\neq1$. In fact, in
such case at temperatures at which the BCS superconducting gaps coincide, the
critical current suddenly increases or decreases. In particular, we thoroughly
discuss the counterintuitively behaviour of the critical current, which
increases by enhancing the temperature of one lead, instead of monotonically
reducing. In this case, we found that the largest jump of the critical current
is obtained for moderate asymmetries, $r\simeq3$. In view of these results, the
discussed behavior can be speculatively proposed as a temperature-based
threshold single-photon detector with photon-counting capabilities, which
operates non-linearly in the non-dissipative channel.
