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$^\dagger$. 

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). 

Some other posts on this site with useful information is https://mathoverflow.net/questions/35655/measure-on-real-grassmannians?noredirect=1&lq=1  and  https://mathoverflow.net/questions/36025/explicit-computations-using-the-haar-measure?noredirect=1&lq=1.

$^\dagger$The surface area of the $n$-sphere is $\frac{2\cdot \pi^{(n+1)/2}}{\Gamma((n+1)/2)}$ and is obtained by integrating the constant 1 with respect to the surface measure on the sphere. So to get the density of the uniform probability measure on the sphere we have to invert that, giving the (constant) density $f(x)=\frac{\Gamma((n+1)/2)}{2 \cdot\pi^{(n+1)/2}}\cdot \mathbb{I}_{(\|x\|=1)}$ and now you can compute the entropy.