  We consider the finite generation property for cohomology of a finite tensor
category C, which requires that the self-extension algebra of the unit
Ext*_C(1,1) is a finitely generated algebra and that, for each object V in C,
the graded extension group Ext*_C(1,V) is a finitely generated module over the
aforementioned algebra. We prove that this cohomological finiteness property is
preserved under duality (with respect to exact module categories) and taking
the Drinfeld center, under suitable restrictions on C. For example, the stated
result holds when C is a braided tensor category of odd Frobenius-Perron
dimension. By applying our general results, we obtain a number of new examples
of finite tensor categories with finitely generated cohomology. In
characteristic 0, we show that dynamical quantum groups at roots of unity have
finitely generated cohomology. We also provide a new class of examples in
finite characteristic which are constructed via infinitesimal group schemes.
