  We provide a formula for $F$-thresholds of a Thom-Sebastiani type polynomial
over a perfect field of prime characteristic. This result extends the formula
for the $F$-pure threshold of a diagonal hypersurface. We also compute the
first test ideal of Thom-Sebastiani type polynomials. Finally, we apply our
result to find hypersurfaces where the log canonical thresholds equals the
$F$-pure thresholds for infinitely many prime numbers.
