As you may know, a landlord in Texas has a warranty of habitability to the tenant under Tex. Prop. Code. §92.052 et seq,, which requires the landlord to make a diligent effort to repair or remedy a condition materially affects the physical health or safety of the tenant.
However, if the tenant's complaints are frivolous, the landlord (or their management company acting as the landlord's agent) can reject/ignore the complaints.
But, unless the tenant's complaints can be defined as stalking (see HERE), then the management cannot get a restraining order or otherwise sue for relief. (There is also applicable rules about harassing or threatening the victim which apply to this too, but I am guessing that the tenant's behavior would not fall under this, either.)
As such, this is simply a bad tenant - an obnoxious tenant. But unless their complaints harbor ill will and are menacing, the best the management company can do is say "no" and hang up.
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