Let \(R\) and \(S\) be standard graded \(k\)-algebras, and \(I\) and \(J\) non-zero, proper homogeneous ideals of \(R\) and \(S\), respectively. Denote by \(P\) the mixed sum of \(I\) and \(J\), that is the sum \(I+J \subseteq T=R\otimes_k S\), where \(I\) and \(J\) are regarded as ideals of \(T\). The authors investigate in the paper several important homological invariants of powers of \(P\), including the projective dimension, regularity and linearity defect, in terms of the information about \(I\) and \(J\). In particular, it is proved in the paper that if \(R\) and \(S\) are polynomial rings over \(k\) and either \(\mathrm{char}k=0\) or \(I\) and \(J\) are monomial ideals then for each \(s\geq1\) there are qualities: \[ \mathrm{depth }T/P^s=\min_{i\in[1,s-1], j\in[1,s]}\{\mathrm{depth }R/I^{s-i}+\mathrm{depth }S/J^{i}+1,\mathrm{ depth }R/I^{s-j+1}+\mathrm{depth }S/J^{i}\} \] and \[ \mathrm{reg }T/P^s=\max_{i\in[1,s-1], j\in[1,s]}\{\mathrm{reg }R/I^{s-i}+\mathrm{reg }S/J^{i}+1, \mathrm{reg }R/I^{s-j+1}+\mathrm{reg }S/J^{i}\}. \]