Tertiary structural stability can be studied using fluorescence spectroscopy from the fluorescence originating from intrinsic fluorophores or from fluorophores dyes covalently linked to the amino acids in catalase. Intrinsic fluorescence can be measured only if there are tryptophans (it is difficult to study with other chromophores) in catalase and provided they are not quenched. With extrinsic fluorophores, one needs to be careful with structural or conformational changes that occur on tagging the dyes to the enzyme.

You may want to look at these publications.

Structural stabilization of [2Fe-2S] ferredoxin from Halobacterium salinarum

AK Bandyopadhyay, G Krishnamoorthy, HM Sonawat

Biochemistry 40 (5), 1284-1292

Structural and Conformational Stability of Horseradish Peroxidase: Effect of Temperature and pH

K. Chattopadhyay and S.  Mazumdar

Biochemistry, (2000) 39(1), 263-270.

There are other publications from these groups too.

Alternatively you could use Circular Dichroism, NMR etc for structural studies.