Distributions on the sphere is studied and used in *directional statistics*, see for instance the text [Directional Statistics](https://www.bookdepository.com/Directional-Statistics-Kanti-V-Mardia/9780471953333?ref=grid-view&qid=1564651041318&sr=1-1) by Kanti Mardia & Peter Jupp. The entropy of the uniform distribution on the $n$-sphere $\mathbb S^n \subset \mathbb R^{n+1}$ is by my calculations $\log\{\frac{2\cdot \pi^{(n+1)/2}}{\Gamma((n+1)/2)}\}$, which simply uses the surface area of the sphere. 

For the non-uniform distribution given, by the sampling process given the radius squared $R^2=\| x\|^2$ will have a [non-central chisquared](https://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution) distribution woth non-centrality parameter $\lambda=t^2$. Then by calculating the conditional density given $R^2=1$ we can find the density on the sphere (details not given) as 
proportional to $\exp(t x_1)$, which by results in chapter 9 of the Mardia&Jupp book proves this is a [von Mises-Fisher](https://en.wikipedia.org/wiki/Von_Mises%E2%80%93Fisher_distribution) distribution. By the same reference, there is a maximum entropy characterization of the von Mises-Fisher distribution with given expectation, [see also](http://www1.maths.leeds.ac.uk/~sta6kvm/reprints/StatisticalDistributions1975.pdf).  With this results it shouldn't be difficult to calculate the entropy directly. And, indeed it is calculated [here](https://escholarship.org/content/qt07h5696q/qt07h5696q.pdf).