Stability of the chromosomes of thermophilic bacteria
The stability and replication of the DNA of thermophiles is an interesting problem, for which there does not appear to be a single unequivocal answer. However the way you express this question is unfortunate: the hydrogen bonds in the DNA of thermophiles (which include some eubacteria as well as archea) are the same as those in any other genomic DNA. The question is, rather, why the strands of the DNA do not dissociate. This is not really my area, but as there have been no other answers I shall summarize what I have learned from the literature. 

A review that was the starting point for this information is:

D.W. Grogan, Hyperthermophiles and the problem of DNA instability Molecular Microbiology (1998) 28(6), 1043–1049.


One obvious possibility would be that the genomes of thermophiles have a high percentage of GC, as GC base-pairs have three hydrogen bonds, whereas AT base-pairs have only two. However, this is NOT the general case.
It has long been known that the melting temperature (the temperature at which 50% strand-separation has occurred) is increased by increased concentration of cations. It has been suggested that this could be a factor, especially in those thermophiles that are also extreme halophiles.
In eukaryotes genomic DNA is condensed into chromatin by the interaction of the basic histone proteins. These proteins are not found in prokaryotes, but different basic molecules, polyamines, have been observed that are more abundant in thermophilic than mesophilic bacteria. As they have been observed to protect DNA from thermal denaturation in vitro, it has been proposed that they may also perform this function in vivo.
Type I topoisomerases that introduce positive supercoils into DNA (‘reverse gyrases’), occur in hyperthermophiles examined but not in mesophiles, and it was suggested that this could stabilize the DNA. However, the Grogan review presents a body of evidence that casts doubt on whether this is, in fact the case.
In addition to strand-separation, the genomes of thermophiles face the problem of increased damage to their DNA. Although, again, the situation is complex, it appears that they have evolved more extensive and active systems of DNA repair to deal with this.