We study the large deviation behavior of lacunary sums \((S_n/n)_{n\in\mathbb{N}}\) with \(S_n:= \sum_{k=1}^nf(a_kU)\), \(n\in\mathbb{N}\), where \(U\) is uniformly distributed on \([0,1]\), \((a_k)_{k\in\mathbb{N}} \) is an Hadamard gap sequence, and \(f\colon \mathbb{R}\to\mathbb{R}\) is a \(1\)-periodic, (Lipschitz-)continuous mapping. In the case of large gaps, we show that the normalized partial sums satisfy a large deviation principle at speed \(n\) and with a good rate function which is the same as in the case of independent and identically distributed random variables \(U_k\), \(k\in\mathbb{N}\), having uniform distribution on \([0,1]\). When the lacunary sequence \((a_k)_{k\in\mathbb{N}}\) is a geometric progression, then we also obtain large deviation principles at speed \(n\), but with a good rate function that is different from the independent case, its form depending in a subtle way on the interplay between the function $f$ and the arithmetic properties of the gap sequence. Our work generalizes some results recently obtained by Aistleitner, Gantert, Kabluchko, Prochno, and Ramanan [Large deviation principles for lacunary sums, preprint, 2020] who initiated this line of research for the case of lacunary trigonometric sums.