To investigate the missing compact star of Supernova 1987A (SN1987A), we analysed the cooling and heating processes of a possible compact star based on the upper limit of observational X-ray luminosity. From the cooling process, we found that a solid quark-cluster star (SQS), having a stiffer equation of state than that of a conventional liquid quark star, has a heat capacity much smaller than a neutron star. The SQS can cool down quickly, naturally explaining the non-detection of a point source in X-ray wavelengths. On the other hand, we considered the heating processes due to magnetospheric activity and possible accretion, and obtained some constraints on the parameters of a possible pulsar. Therefore, we concluded that an SQS can explain the observational limit in a confident parameter space. With a short period and a strong magnetic field (or with a long period and weak field), a pulsar would have a luminosity higher than the observational limit if the optical depth is not large enough to hide the compact star. As possible central compact objects, the parameters constrained for a pulsar can be tested for SN1987A with advanced facilities in the future.