Let \(R\) be a commutative noetherian ring, \(\mathfrak{a}\) an ideal of \(R\), and \(M\) and \(N\) two \(R\)-modules. Herzog defined the \(i\)th generalized local cohomology module of \(M\) and \(N\) with respect to \(\mathfrak{a}\) as follows: \[H^{i}_{\mathfrak{a}}(M,N) = \underset{n\in \mathbb{N}}\varinjlim \operatorname{Ext}^{i}_{R}(M/\mathfrak{a}^{n}M,N)\] On the other hand, \textit{R. Hartshorne} [Invent. Math. 9, 145--164 (1970; Zbl 0196.24301)] defined \(M\) to be \(\mathfrak{a}\)-cofinite if \(\operatorname{Supp}_{R}(M) \subseteq V(\mathfrak{a})\) and \(\operatorname{Ext}^{i}_{R}(R/\mathfrak{a},M)\) is a finitely generated \(R\)-module for every \(i\geq 0\). Moreover, recall that \[\operatorname{cd}(\mathfrak{a},M)=\sup\{i \geq 0\mid H^{i}_{\mathfrak{a}}(M)\neq 0\},\] and \[\operatorname{q}(\mathfrak{a},M)=\sup\{i \geq 0\mid H^{i}_{\mathfrak{a}}(M) \text{ is not artinian}\}.\] Now suppose that \(M\) and \(N\) are finitely generated. The authors prove that if \(\operatorname{cd}(\mathfrak{a},R)\leq 1\) and \(M\) has finite projective dimension, then \(H^{i}_{\mathfrak{a}}(M,N)\) is \(\mathfrak{a}\)-cofinite for every \(i\geq 0\). Furthermore, they show that if \(\operatorname{q}(\mathfrak{a},R)\leq 1\) and \(M\) has finite projective dimension, then the Bass numbers of \(H^{i}_{\mathfrak{a}}(M,N)\) are all finite for every \(i\geq 0\). Finally, they characterize the greatest integer \(i\) for which \(H^{i}_{\mathfrak{a}}(M,N)\) is not artinian and \(\mathfrak{a}\)-cofinite.